The Spaniards saw off Argentinian pair Maximo Gonzalez and Andres Molteni in straight sets 7-6 (6-4) 6-4 in one hour and 49 minutes on Saturday.
Alcaraz, who began the day with a victory in his singles debut, made a nervy start as he dropped his first serve, but Nadal soon helped to settle the nerves.
They got the vital break in the next game, though there was little to separate the teams after that with a tie-break necessary to put Alcaraz and Nadal in front.
Gonzalez and Molteni rallied in the second set, racing into a 3-0 lead, but could not prevent a fight back as Alcaraz and Nadal won the next three games.
After another vital break for the Spaniards at 4-4, Nadal then served out the match to set up a meeting with either Dutch pair Tallon Griekspoor and Wesley Koolhof or Marton Fucsovics and Fabian Marozsan of Hungary.
Data Debrief: Living up to expectations
It is the match-up that everybody was waiting for at the Olympics, and despite a sluggish start, Alcaraz and Nadal soon turned on the style.
They hit 17 winners, and even though Alcaraz's first service game was not his usual best, they went on to win 46 of their 64 service points (72%).
The reigning French Open champion was pushed all the way in his straight-sets victory, getting a 6-1 7-6 (7-3) win in one hour and 48 minutes.
It looked like it was going to be another routine match for the Spaniard at Roland-Garros as he raced into a 3-0 lead in the first set, and even after Griekspoor pulled one back, Alcaraz's aggressive approach saw him take the lead.
The Dutchman found his fight in the second, though, with the players matching each other stride for stride. Griekspoor threatened to take the set with a break, but Alcaraz successfully defended it before forcing a tie-break.
Alcaraz then turned on the style when he needed it most, winning the last four points in a row to book his place in the next round, where he will face Roman Safiullin.
Data Debrief: Alcaraz gets the job done
Unlike his opening round, when he cruised through with a 6-3 6-1 victory over Lebanon's Hady Habib, Alcaraz was made to work hard for this victory.
Despite the stern test, Alcaraz did not drop a single service game, successfully saving two break points.
He is back in action on Tuesday when he will once again pair up with Rafael Nadal in the men's doubles.
The Spaniard, who overcame fitness concerns in his win over Tallon Griekspoor in the last round, dominated on Court Suzanne Lenglen to win 6-4 6-2 in an hour and a half.
Alcaraz made his intent clear from the start, breaking Safiullin twice in the opening three games to race into the lead.
The Russian regained his footing shortly after to make it a more even affair, but he managed just one break as Alcaraz cruised through the first set.
The 21-year-old made a similar start in the second and twice threatened to take a three-game advantage over his opponent, but Safiullin successfully defended two break points to make it 3-2.
He could not do the same on his next serve though, and Alcaraz won the last three games in a row to set up a meeting with the United States' Tommy Paul in the last eight.
Data Debrief: Just no stopping Alcaraz
Alcaraz has reached the quarter-finals in every high-profile tournament he has entered, winning the last two such events at the French Open (at Roland-Garros) and Wimbledon.
He is also the youngest player to make the men's singles quarter-finals at the Olympics since Novak Djokovic in Beijing 2008.
The Spaniard – who captured his fourth grand slam title at Wimbledon earlier this month – needed just one hour and 12 minutes to overcome Habib, who is ranked 275th in the world by the ATP.
Alcaraz had to save break point in the very first game but immediately set about putting Habib under pressure from there, getting a quickfire break to take control of the opener.
Habib hit just three winners throughout the first set, with 16 unforced errors costing him as Alcaraz served with supreme confidence, only dropping one point in a span of six games on his own serve.
Another break in the first game of the second set had Alcaraz truly in charge, and he gave nothing away from then on as he eased to victory.
He will face Britain's Cameron Norrie or the Netherlands' Tallon Griekspoor in the second round, with the duo yet to face off due to rain suspending play on the outside courts.
Data Debrief: Alcaraz goes from strength to strength
Returning to the scene of last month's French Open triumph, Alcaraz was barely troubled as he improved his record at clay-court events to 12-3 for the year.
He was far too good for Habib on Saturday, slamming 26 winners to his opponents' five and causing problems whenever he was able to get to the net, where he won 13 of 16 points on approach (81%).
He will return to the court to partner Rafael Nadal in the doubles later on Sunday, versus Argentine duo Maximo Gonzalez and Andres Molteni.
The Spaniard won in straight sets, 6-1 7-6 (7-3), but had to take a medical timeout partway through the second to receive treatment.
After a comfortable first set, Alcaraz was then pushed all the way to make it into the third round in the men's singles, while he is also competing in the men's doubles with Rafael Nadal.
Alcaraz was quick to downplay the injury, noting how he has not let it affect him in recent tournaments, including his Wimbledon win earlier this month.
"It's a problem I've been having since the last few tournaments and I have to pay special attention to that area," Alcaraz said after the match.
"Tennis players have a very demanding schedule, you play more matches and problems arise, but the good thing is that I know how to deal with those discomforts and how to recover.
"It's an area where I've been feeling it for several months, but I've been able to play perfectly in the last few tournaments and this one will be no different."
Alcaraz and Nadal will play Netherlands' Griekspoor and Wesley Koolhof in doubles on Tuesday before the 21-year-old's third-round meeting with Roman Safiullin later in the week.
Kerber, who won singles titles at the Australian Open, US Open and Wimbledon between 2016 and 2018, returned to the court after an 18-month hiatus earlier this year.
She had missed the entire 2023 season after announcing her pregnancy in August 2022, giving birth to a daughter the following February.
The 36-year-old has fond memories of the Olympics, having claimed silver at Rio 2016, and she sees this year's tournament at Roland-Garros as a fitting place to end her career.
"The finish line. Before the Olympics begin, I can already say that I will never forget Paris 2024, because it will be my last professional tournament as a tennis player," Kerber wrote on Instagram on Thursday.
"Whereas this might actually be the right decision, it will never feel that way. Simply because I love the sport with all my heart and I'm thankful for the memories and opportunities it has given me.
"The Olympics I've participated in so far have been more than just competitions as they represent different chapters of my life as a tennis player: the climb, the peak… and now, the finish line.
"Paris 2024 will mark the finish line of the most incredible journey I could have ever dreamed of, growing up with a racket in my hand.
"There are many more things I want to say and people to thank, which I will do once I completed my last match… but for now, I will take the time and soak up every second of this final episode on court."
Spain are aiming to become the first side to win Olympic gold after lifting the Women's World Cup, having defeated England in the final of that tournament last year.
Ballon d'Or winner Bonmati levelled after Aoba Fujino's stunning 13th-minute free-kick opener in Nantes, before the Spain midfielder teed up Mariona Caldentey's winner 16 minutes from time.
That result saw Montserrat Tome's side seize top spot in Group C ahead of Brazil, who edged past Nigeria 1-0 thanks to Gabi Nunes' first-half finish.
Elsewhere in Group A, defending champions Canada put off-field issues to one side by overcoming New Zealand 2-1 thanks to goals from Cloe Lacasse and Evelyne Viens in Saint-Etienne.
Canada's integrity had been questioned ahead of the game in the wake of a spying scandal, with assistant Andy Spence serving as head coach after Bev Priestman volunteered to sit out.
Priestman, two members of her staff and the Canadian Olympic Committee are being investigated by FIFA's disciplinary committee after a drone was spotted flying over New Zealand's training on Monday.
"Obviously the timing was quite terrible," said Canada captain Jessie Fleming of the drone incident after her side battled back following Mackenzie Barry's early opener.
"But I think for us, we've just leaned into each other as a player group and we really just want to focus on being here and playing."
Hosts France seemed on course for a routine victory in the other Group A game after Marie-Antoinette Katoto's first-half double and Kadidiatou Diani's goal against Colombia.
However, Catalina Usme pulled one back from the spot and Manuela Pavi further reduced the deficit nine minutes later, only for Mayra Ramirez's late red card to help France hold out for a 3-2 victory.
Meanwhile, Germany swept aside Australia 3-0 in Group B, inflicting the Matildas' joint-heaviest defeat at the Olympics, along with a loss against the same opponent by the same scoreline, at Sydney 2000.
Efforts from Marina Hegering, Lea Schuller and Jule Brand sealed the victory for the DFB-Frauenteam against the Tokyo 2020 semi-finalists.
After being found guilty of using a drone to film opponents' training at this and past tournaments, the Tokyo champions had six points deducted, while head coach Bev Priestman has been banned from all football-related activity for one year.
The Canadian Soccer Association (CSA) are considering an appeal against the decision, but those controversies were put to one side when the team took to the field.
Starting the day bottom on -3 points, Canada, led by Andy Spence, had to come from behind to beat France after Marie-Antoinette Katoto put Les Bleues ahead just before half-time.
Jessie Fleming equalised from close range in the 58th minute, but Canada fought back, with Gilles getting some help from the woodwork to secure the win.
Meanwhile, Colombia's comfortable 2-0 victory over New Zealand means they now lead Group A after two games thanks to their goal difference.
Elsewhere, in Group C, Spain booked their place in the quarter-finals with a game to spare after a 1-0 win over Nigeria.
Alexia Putellas' long-range strike maintained La Roja's perfect start to the tournament as they bid to become the first team to win Olympic gold after lifting the Women's World Cup.
In the other group game, Japan ruined the party for Marta, who made her 200th appearance for Brazil in a shock 2-1 defeat.
Despite Mina Tanaka's penalty miss in the first half, Japan launched a comeback after Jheniffer's opener, with Saki Kumagai and Momoko Tanikawa both scoring in stoppage time to move second.
The game of the day came in Group B though, as Australia edged out Zambia in an 11-goal thriller.
Barbra Banda's hat-trick gave the minnows a 5-2 lead, but they could not hold onto it as Michelle Heyman made it 6-5 in the 90th minute to snatch a win.
Murray confirmed this week that the tennis tournament at Roland Garros, which begins on Saturday, will be his last before he hangs up his racket.
The three-time grand slam champion has endured an injury-hit few years and was unable to make a final appearance in the singles draw at Wimbledon this month, though he did team up with brother Jamie for one last match on Centre Court – a loss to John Peers and Rinky Hijikata.
Murray is the only men's player to have won two Olympic golds ahead of his Paris campaign, but Cavendish – who made history by claiming a 35th Tour de France stage win earlier this month – says his legacy goes beyond his results.
"What Andy has showed, whatever you do, just do it. It's not for anyone else to say what you can and can't do – just do what you love," Cavendish said.
"Aside from his incredible achievements on the court, what Andy has stood for off the court, is his fight for equality between male and female sportspeople especially, and his vocal criticism of people who criticise him and what he does.
"Absolute love of a sport and carrying the hopes of a nation for so long – he's an incredible person.
"Whatever he does, you know he's going to do it with passion and I wish him the best."
The 24-time major winner took just 53 minutes to wrap up a comprehensive 6-0 6-1 rout on Court Philippe-Chatrier, to set up a potential second-round clash with Rafael Nadal.
Djokovic won bronze on his Olympic debut in 2008, but missed out on a medal after finishing fourth in 2012 and 2020, while he was beaten in round one in Rio by eventual silver medallist Juan Martin del Potro eight years ago.
However, there was no danger of a repeat of the latter this time around. The Serb took just 24 minutes to wrap up a dominant first set, in which he bageled his helpless opponent.
Ebden remained in positive spirits despite his ordeal. At one stage, he offered his racquet to the crowd, who gave him a wonderful reception when he won his first game of the contest at the 11th attempt.
Nevertheless, it only briefly delayed the inevitable, with Djokovic cruising to the finish line.
Data Debrief: Dominant Djokovic soars into second round
There was no messing around from the 24-time major winner on Saturday, as he was done and dusted in just under an hour at Roland-Garros.
He converted six of his nine break-point opportunities, while winning 14 of 19 net points (74%).
The identity of his second-round opponent is still to be decided, but we could well see a blockbuster clash with Nadal.
Billed as their 'last dance', the great rivals faced off for a record 60th time in the Open Era, with Djokovic coming out on top with a 6-1 6-4 straight-sets win.
It was not quite so straightforward for the Serb though, as Nadal showed his fight in the second set, fighting from 4-0 down to tie it at 4-4 before Djokovic wrestled back control to win.
The 37-year-old is the first player to win 15 men's singles matches at the Olympics since tennis returned to the summer programme in 1988, but he is still searching for an elusive first gold medal.
And despite a slightly sloppy finish to the second set, Djokovic was pleased with his overall performance.
"At 6-1, 4-0, I got a little bit too comfortable," Djokovic said. "At 4-1, I played a little sloppy service game, and you can't give any chances to Nadal because he is going to use them and comeback.
"Especially on this court. The crowd getting involved, but it was a crucial game at 4-4, trying to break his serve and then serve against the wind with new balls.
"It was a very close encounter, but I am very pleased with the way I played."
The win was also Djokovic's 31st against Nadal, the most match wins against a single opponent by anyone in the Open Era.
With this potentially the last-ever meeting, Djokovic reflected on just how far they had come since first facing off at Roland-Garros in the French Open quarter-finals in 2006.
"Back in 2006, I don't think we would have thought we would be playing against each other on the same court at the Olympics," he added.
"So, I think we will appreciate this match very much. I think our rivalry and I think for the sport itself.
"I think there was a lot of attention and interest directed from sports towards us. It is unfortunate for him that he wasn't at his best, but I did everything I could to make him feel uncomfortable."
Djokovic will face Germany’s Dominik Koepfer in the next round.
Thursday's men's singles draw saw Djokovic paired with Australia's Matthew Ebden in the first round, with Nadal taking on Marton Fucsovics.
The legendary duo – who have won 46 grand slam singles titles between them – will meet in the second round if they both advance through their openers this weekend.
Djokovic and Nadal have previously faced off on 59 occasions, with the Serbian winning 30 of those contests but losing four of the last six, the most recent being a Nadal victory in four sets in the 2022 French Open quarter-finals.
Nadal's participation at the Games was thrown into doubt when his coach Carlos Moya said he had suffered a "setback" on Thursday and required 48 hours of rest.
If Nadal – who is widely expected to retire later this year – does make the tournament, Djokovic is looking forward to the prospect of facing him for a 60th time.
"Obviously he hasn't played much. His ranking dropped, so there was always going to be that possibility of me meeting him in an early round and there we go," Djokovic said.
"It's going to be a spectacle, definitely, if we get to meet. We will spread some fireworks on the court, like the good old times.
"I hope we get to meet because it will probably be one last dance for the both of us."
Djokovic is set to appear at his fifth Olympics in the French capital, the joint-most by any male player since the sport returned to the Games in 1988 (with Yen Hsun Lu).
He is the player with the most matches played (19) and joint-most matches won (13, level with Roger Federer) at the Olympics since 1988.
However, his best result at the Games remains a Bronze medal at Beijing 2008, when he was beaten by eventual gold medallist Nadal in the semi-finals.
Djokovic lost the bronze medal match to Pablo Carreno Busta at the delayed Tokyo Games three years ago, also losing to Juan Martin del Potro at both London 2012 and Rio 2016.
Miguel Gutierrez also struck for La Roja, who made it two wins from two in Group C at Stade de Bordeaux, where their opponents played over half of the game with 10 players following Edison Azcona's red card.
Santi Denia's side opened their campaign with a 2-1 victory over Uzbekistan at Parc des Princes on Wednesday, and the first goal of this contest was handed to them on a plate in the 24th minute.
Fermin ruthlessly pounced on Enrique Boesl's tame pass out of goal, with the Barcelona midfielder calmly slotting home.
The Dominican Republic levelled 13 minutes later through Angel Montes de Oca's free header, but they were reduced to 10 men just before half-time with Azcona dismissed for violent conduct after lashing out at Pau Cubarsi.
Spain capitalised on their numerical advantage through Alex Baena's deflected strike in the 55th minute before substitute Gutierrez sealed the points 15 minutes later by tapping into an empty net, with the goal eventually standing following a VAR review.
Data Debrief: La Roja roar into quarter-finals
Spain dominated large periods of the contest with 21 shots to their opponents' five and, playing the entire second half with an extra player, their superiority eventually told.
Silver medallists in Tokyo three years ago, La Roja have progressed from the group stage of consecutive Games for the first time since a run of three straight qualifications between 1992 and 2000.
The United States will go in search of a fifth straight men's basketball gold in the French capital, where they open their Group C campaign against Serbia on Sunday.
After that meeting with Nikola Jokic and company, the star-studded USA team featuring Durant, LeBron James, Stephen Curry, Joel Embiid and a host of other NBA stars take on South Sudan and Puerto Rico.
With Steve Kerr having so much talent at his disposal, the roster has been compared to that of the 1992 'Dream Team', who took gold in Barcelona behind the efforts of Michael Jordan, Magic Johnson, Larry Bird, Scottie Pippen and other great names.
However, at a press conference ahead of Team USA's 2024 opener, Curry reminded reporters that the class of 1992 only faced nine other active NBA players.
Durant is expecting a much tougher assignment this year, responding to Curry by saying: "Everybody is a contender. Everyone is here for a reason.
"We're playing against 65 or 64 NBA players in Paris. So you know they're the best in the world."
Asked to identify the main players to watch from other nations, the Phoenix Suns forward said: "Giannis [Antetokounmpo] or Jokic."
While Durant adopted a cautious tone, Golden State Warriors star Curry was more bullish regarding the USA's gold medal hopes.
Quizzed on who he was most excited about facing, Curry simply said: "Whoever's in the gold medal game."
Peaty was bidding to become just the second man after Michael Phelps to win the same swimming event at three consecutive Olympics.
However, he had to settle for a tie for silver with America's Nic Fink, after Italy's Nicolo Martinenghi snatched a shock win by beating them to the wall by 0.02 seconds.
Peaty recently took a break from the sport to cope with mental health struggles, which came to a head when he broke his foot in 2022, ending his eight-year dominance over 100m.
Having made his comeback with a bronze at this year's World Championships, a tearful Peaty explained why he was not disappointed about falling short of the top prize.
"It's been a very long way back. I look at the score there, you never think in your wildest dreams it would be 59 [seconds] to win it," he told the BBC.
"I gave my absolute all and executed it as well as I could. It's not about the end goal, it's all about the process. It doesn't matter what the end time says on the board, in my heart, I've already won.
"I'm not crying because I came second, I'm crying because it took so much to get here. It's incredibly hard to win it once, and to win it again, and again and trying to find new ways to do it.
"Everything I have done to this point has happened for a reason. I'm so happy I can race the best in the world and get joint second.
"In my heart, I've won. These are happy tears. I said to myself to give my absolute best every single day and I have. You cannot be upset about that."
Peaty will have another chance to go for gold in Paris when he competes in the 4x100m relay.
Les Bleus made it two wins from two thanks to the right-back's header, though they rode their luck against a spirited Guinea side, who had two first-half goals disallowed.
Amadou Diawara forced Guillaume Restes into a superb fingertip save just minutes into the tie before Jean-Philippe Mateta wasted a hat-trick of chances to put the hosts ahead.
A cheeky backheel dribbled wide, and a thumping header missed the target on either side of Soumaila Sylla's smothering save.
Naby Keita clinically picked out the bottom corner from an offside position just before half-time, with Abdoulaye Toure also seeing a goal ruled out moments later after meeting a free-kick.
France came to life in the second half and gave a warning sign when Loic Bade struck the post from a tight angle.
In the end, Sildillia proved the difference as he was left unmarked to meet Michael Olise's cross, and Sylla could not get to his thumping header.
France are through to the next round, and top Group A with six points, while Guinea sit bottom of the group having lost both of their matches so far.
Data Debrief: Les Bleus leave it late
For much of the first half in this game, France were sluggish and could have easily gone into half-time 2-0 down if it was not for two tight offside calls.
In the end, Thierry Henry's side showed their quality, with Olise proving the key man. He created six chances, more than anyone else on the pitch, with one of those resulting in an assist.
It is the first time France have reached the quarter-finals at the Olympics since 1996, as they looked to end a 40-year wait for a gold medal in the event.
The American cruised through in just 57 minutes on Court Philippe Chatrier, winning 6-3 6-0 in the evening session.
Gauff raced into a 3-0 lead in the first set and did well to hold off the Australian's fightback after trading breaks for the next four games. She soon regained her advantage at 5-3 though, before winning the final game to love to take the lead.
Gauff eased through the second set with little fuss and Tomljanovic had no answers as she finished the game on the wrong end of a bagel.
Next up for Gauff is Argentina's Maria Lourdes Carle, who routed Tatjana Maria 6-0 6-0 on Sunday.
Data Debrief: Gauff gets off to perfect start
Gauff missed the Tokyo 2021 Olympics through illness, and she could not have asked for a better start in Paris after winning both her doubles and singles matches.
The 20-year-old won the final eight games in a row en route to her dominant win, breaking Tomljanovic's serve six times throughout the match.
The American took just one hour and 26 minutes to brush Carle aside 6-1 6-1 on Court Suzanne Lenglen on Monday.
Gauff stormed into a 5-0 lead in the first set before the Argentinian found her fight – she held break point three times in the final game, but the 20-year-old finally converted on her fourth set point.
It was much the same in the second, with Gauff cruising through. Though she lost her serve in the penultimate game, she made quick work in the last to set up a meeting with Donna Vekic of Croatia.
Data Debrief: Gauff continues to shine
Gauff remains undefeated in her Olympics debut, staying active in all three disciplines as she is also competing in the women's doubles and mixed doubles.
She followed up a dominant win on Sunday with an equally impressive performance as she converted eight break points on Monday while only dropping serve once.
Despite a slow start by the hosts, which saw them trailing early in the second quarter, Wembanyama helped to steady their nerves in Lille.
The NBA Rookie of the Year last season came away with 19 points, nine rebounds, four steals and three blocked shots as he inspired his nation in front of a sell-out crowd.
France did not fully seal the victory until late on though, with Nicolas Batum's 3-pointer putting them up by 10 with less than a minute and a half remaining before Wembanyama's dunk made sure of the points in the next possession.
"I knew it was going to be a good day," he said.
"You can't underestimate the power of the crowd, the home crowd. I think it's really going to be the sixth man for this whole tournament."
Meanwhile, team-mate Evan Fournier was quick to heap praise on the 20-year-old after the game.
"I think he did pretty damn good," he said. "He can do so many things that sometimes it's like a burden because he doesn't focus on just one thing."
France will face Japan in their second game of Group B on Tuesday before their final match against Germany.
Murray is a two-time Olympic champion, having won men's singles gold medals in 2012 - the year he also won a silver medal in the mixed doubles alongside Laura Robson - and 2016.
The three-time grand slam winner is eligible to compete in the singles in Paris, where two places are reserved for former Olympic or major champions.
Although, he appears set to miss out on the doubles, regardless, with Great Britain set to send Joe Salisbury and Neal Skupski.
The tournament could prove to be the final one in the career of Murray, who revealed in February that he does not "plan on playing much past this summer".
He suffered a third successive first-round defeat in Stuttgart, having also fallen at that stage in the French Open, which will see him drop out of the world's top 100 for the first time in two years.
And Murray is uncertain if he will feature at the quadrennial event.
"I need to see what happens with the Olympics," he told reporters after losing to Marcos Giron in Stuttgart.
"I'm not 100 per cent sure what the situation is there with the doubles yet and whether or not I will play if I just get in the singles. I don't know.
"My body didn't feel great playing on the clay in the last month or so. I had quite a few issues with my back, so I don't know if I would go just for singles. I need to wait a little bit and see on that."
