Spaniard Muguruza was due to face Anastasia Pavlyuchenkova until the Russian withdrew due to an abdominal issue, giving Tig a reprieve and the chance of an upset.
But Muguruza won all but three games in a comprehensive 6-1 6-2 victory that required little more than an hour on court as the former world number one went to 22-6 for the season.
Conditions were not great as drizzle persisted for much of the match, but Muguruza was grateful to progress after a generally testing day.
"It was a weird day," the 12th seed said afterwards. "I woke up [and the weather was] sunny, had an opponent. Then it turned out I had another opponent and it was raining the whole match. We had a little bit of everything!
"But, Rome, it's such a familiar tournament. I played here so many times, I've had nice matches. I love this clay, the red clay. It's familiar to me. I'm just looking forward to getting as many matches as possible."
Bencic will not be joining her in the next round, as she became the highest seed to be eliminated on the day in a 6-3 6-4 defeat to Kristina Mladenovic.
The two played each other under two weeks ago in the first round of the Madrid Open, with Bencic victorious, but Mladenovic got her revenge as she capitalised on her opponent's seven double faults.
Like Tig, Sloane Stephens was also competing as a lucky loser as she came up against compatriot Madison Keys, though the outcome was no different.
Keys claimed only her second win in six meetings with Stephens but had to come from behind as she eventually clinched it 4-6 6-2 7-5 after two-and-a-half hours on court to set up a clash with 15th seed Iga Swiatek.
Elise Mertens, the 14th seed, was also dumped out of the competition, losing out 4-6 6-2 6-3 to Veronika Kudermetova, whom she had beaten in straight sets in their three WTA Tour contests before this one.
Veteran Kazakh Yaroslava Shvedova, ranked 665 in the world, awaits top seed Ashleigh Barty after the 33-year-old showed character to fight back and beat Martina Trevisan 0-6 7-6 (7-4) 7-6 (8-6), while Nadia Podoroska set up a second-round clash with Serena Williams after seeing off Laura Siegemund 2-6 7-6 (7-3) 6-1.
Four-time major winner Osaka has not played competitively since her 2022 Australian Open third-round exit to Amanda Anisimova in January.
The Japanese, whose ranking has dropped to 78th, had been handed a wild card for Indian Wells but was upgraded to the main draw after Camila Giorgi's withdrawal.
The Indian Wells draw was made on Monday with Osaka to face Stephens who won her first WTA title since 2018 with last month's Abierto Zapopan crown.
The top seeds all have been waived through to the second round with top seeds Barbora Krejcikova and Aryna Sabalenka book-ending the draw.
Osaka was in action for the first time since a shock early exit at the Australian Open in January, but she ultimately had too much for Stephens in a 3-6 6-1 6-2 success.
Stephens initially looked the sharper of the two former major champions in the second-round clash, impressively taking a first-set lead, but Osaka found her groove after a slow start to level the contest.
Osaka then had to save three break points at 2-0 down in the decider, and that appeared to inspire another surge as she rattled off six successive games to roar to an emphatic win.
Shelby Rogers got past Nuria Parrizas-Dias in almost three hours, winning 6-1 5-7 7-6 (7-3), and last week's Monterrey Open runner-up Camila Osorio retired down 6-4 5-0 against Aliaksandra Sasnovich.
Yulia Putintseva got past Ashlyn Krueger 6-3 6-2 and Ukraine's Marta Kostyuk won in three hours and nine minutes over Maryna Zanevska 6-7 (5-7) 7-6 (8-6) 7-5.
Last month's Abierto Zapopan runner-up Marie Bouzkova eased past China's Wang Qiang 6-3 7-6 (7-5), while American Amanda Anisimova defeated compatriot Emma Navarro 6-2 6-2.
Ostapenko looked to be cruising after winning the opener, but Kanepi levelled in the second despite the Estonian watching five set points come and go.
Kanepi then romped to victory with a stunning final set to defeat Ostapenko 4-6 6-4 6-0 and keep her hopes of winning her home tournament alive.
In second-round action, Beatriz Haddad Maia is through to the quarter-finals after a 6-1 7-5 success over Linda Noskova.
After Haddad Maia cruised to a one-set lead, the second was a much tighter affair before the Brazilian's class showed as she broke Noskova and then held serve to earn a straight-sets win.
Second seed Belinda Bencic also secured her place in the final eight with a 6-4 6-7 (2-7) 6-3 triumph over Katie Boulter.
At the Parma Ladies Open, top seed Maria Sakkari came from behind to beat Arantxa Rus and seal her quarter-final spot.
Rus saved five out of five break points as she won the opening set 6-3 to give her hope of an upset.
Those hopes were soon dashed though as world number seven Sakkari broke twice to take the second set 6-2, before clinching the match in the third to tee up a clash with Maryna Zanevska.
World number 33 Irina-Camelia Begu made it 12 consecutive wins on clay by overcoming Matilde Paoletti in straight sets and will face fellow Romanian Ana Bogdan in the next round after she beat Anna Karolina Schmiedlova 6-2 3-6 6-3.
Fourth seed Sloane Stephens saw her tournament come to an end after she was defeated by Danka Kovinic.
Having endured an injury-hit season to date, world number eight Pliskova made the perfect start to her campaign in France with a routine 6-4 6-2 win over Ukrainian 19-year-old Marta Kostyuk.
Kerber, who is only missing a victory at the French Open – which begins later this month – to complete a Career Grand Slam, overcame teenager Diane Parry 6-3 6-4 in her opener, setting up a round-of-16 clash with world number 45 Aliaksandra Sasnovich.
Some of the other big names participating in Strasbourg, however, were not so lucky, with 5th and 6th seeds Shuai Zhang and Sloane Stephens crashing to first-round exits.
Shuai went down 6-2 6-3 to Belgium's Maryna Zanevska and Stephens capitulated to a 2-6 6-4 6-1 loss to Nefisa Berberovic, while Heather Watson set up a last-16 meeting with eighth seed Magda Linette by beating Katharina Hobgarski 6-4 6-1.
Meanwhile, Garbine Muguruza made a winning start to her campaign at the Grand Prix Sar La Princesse Lalla Meryem in Morocco, as the world number 10 builds towards a tilt at French Open glory having triumphed at Roland Garros back in 2016.
Muguruza had few difficulties in beating Anna Danilina 6-3 6-2, and will face Martina Trevisan in the round-of-16 after the Italian beat China's You Xiadoi 6-0 6-4.
Third seed Nuria Parrizas-Diaz had to come through a more difficult challenge in Rabat, overcoming Carol Zhao 7-6 (7-4) 1-6 6-2, while eighth seed Anna Kalinskaya was the highest-ranked player to fall at the first hurdle, going down 6-3 6-7 (5-7) 6-3 to Lucia Bronzetti.
The world number 168, who had not played since the Madrid Open in April, marked her return from injury with a run to the Nottingham Open semi-finals just over a week ago.
Last year, Raducanu underwent wrist and ankle surgery that, at one point, confined her to a mobility scooter, but she has prioritised fitness over competition during the 2024 season.
Having missed the French Open to focus on preparing for the grass-court season, she has also opted not to participate for Great Britain in the forthcoming Olympics.
The 21-year-old is now stepping up her preparations for Wimbledon, which continue when she faces Sloane Stephens in Eastbourne on Tuesday.
And Raducanu will enter the contest against her fellow former Flushing Meadows champion with a real spring in her step.
"I really feel like I'm in a lot better spirits on and off the court," she told BBC TV. "It's a nice feeling for me because it hasn't always been the case.
"I'm very happy to have rekindled the love for tennis and just enjoying the challenges that come with it and not getting so down about it, just trying a way to go around.
"I'm just really grateful to have this feeling again because it's something that I feel like I've been missing in a way for the last few years.
"I haven't felt this good about my tennis and excited about it and passionate for a long, long time."
Raducanu, who is set to drop out of the WTA's top 100, withdrew from her first-round meeting with Viktoriya Tomova on Wednesday due to a hand injury.
The 20-year-old Briton has endured a difficult season, with her best performance coming at the Indian Wells Open, where she lost to world number one Iga Swiatek in the last 16.
Fellow youngster Fernandez has also struggled, and she came unstuck against world number 194 Mirra Andreeva, who progressed 6-3 6-4.
The big shock came as Jaqueline Cristian, ranked 525th in the singles rankings, defeated 2017 US Open champion Sloane Stephens.
Having taken the first set, Stephens looked set to book her place in round two, only for Romanian Cristian to prevail 5-7 6-4 6-4.
Alize Cornet overcame world number 39 Elena-Gabriela Ruse 6-7 (6-8) 6-4 6-3, while Camila Giorgi was forced to retire from her match with Mayar Sherif before the start of a deciding third set, and Amanda Anisimova was another top-50 casualty.
Rebeka Masarova, Nuria Parrizas Diaz, Marina Bassols Ribera and Irene Burillo Escorihuela carried the flag for Spain into the next round.
Eugenie Bouchard got the better of Dayana Yastremska as her return to top-level tennis continues.
Raducanu had to come from a set down to beat Linda Fruhvirtova 4-6 6-4 6-2, while top seed Gauff had an easier time of it as she dismissed Tatjana Maria 6-4 6-1 on another day interrupted by the rain.
Second seed Sloane Stephens was only able to get through three games before play was suspended in her match against Rebeka Masarova, while fifth seed Wang Xiyu was 5-3 down in the first set against Karolina Muchova when the rain came.
Fourth seed Bernarda Pera is out after her match with Viktoria Kuzmova resumed having being washed out on Monday, and the American was unable to prevent a 6-4 6-4 defeat, with Kuzmova advancing to face Raducanu in the last 16.
The only two other matches completed on Tuesday saw Elena-Gabriela Ruse beat Erin Routliffe in three sets, and Anna Blinkova overcome Elisabetta Cocciaretto, also in three.
At the Adelaide International, Zheng Qinwen earned a hard-fought victory against number six seed Anett Kontaveit, eventually winning 6-1 4-6 7-6 (9-7).
Eighth seed Ekaterina Alexandrova also crashed out despite winning the first set against Marketa Vondrousova, going on to lose 4-6 6-3 6-2, though seventh seed Jelena Ostapenko is through after beating Karolina Pliskova 6-1 6-3.
In a contest that lasted over two-and-a-half hours despite only two sets being played, Victoria Azarenka beat Anhelina Kalinina 7-6 (11-9) 7-6 (7-5), and will face Zheng next, while Sorana Cirstea will go up against top seed Ons Jabeur in the last 16 after beating Viktorija Golubic in straight sets.
Despite an early scare, Raducanu cruised through, winning 6-4 6-0 in just one hour 17 minutes.
Raducanu and Stephens traded blows early on in the first set, with both getting double breaks to stay neck-and-neck before the Briton edged it in the final game by breaking the American's serve once more.
She then brushed Stephens aside in just 25 minutes for the second without dropping a single game, to set up a meeting with Jessica Pegula in the next round.
Data Debrief: Raducanu in control
Raducanu finished off the match in style, winning the last nine games in a row, and broke Sloane's serve six times.
The 21-year-old finished with 23 winners to just 13 unforced errors to tee up a marquee match-up in the second round.
The Briton, who won the US Open in 2021, went down 6-3 3-6 6-2 in a genuine tug-of-war against the 31st-ranked Canadian.
Andreescu sent down seven aces for the match, including four in the first set along with 12 winners as she claimed an early lead.
Raducanu responded with greater aggression, securing the crucial break in the eighth game, before serving out to level the contest.
But Andreescu showed composure and quality in the big points in the final set, winning the final four games to round out the victory, sealing a second-round clash with seventh seed Maria Sakkari.
Andreescu finished with 32 winners compared to Raducanu's 20, while the Briton committed 40 unforced errors.
World number 42 Shelby Rogers edged out USA compatriot Sloane Stephens in just under two hours, winning 6-4 3-6 6-2. Rogers will take on second seed Aryna Sabalenka, who is arguably the tournament favourite after top seed Iga Swiatek's withdrawal, in the next round.
Rogers was one of five Americans to progress, including 2020 Australian Open champion Sofia Kenin who defeated Australia's Storm Hunter 6-0 7-6 (7-5).
World number 35 Amanda Anisimova retired down 7-6 (7-5) 5-2 to countrywoman Madison Brengle.
Leylah Fernandez, the player whom Raducanu beat in the 2021 US Open final, advanced into the second round with a 6-4 6-3 victory over Lesia Tsurenko in 88 minutes.
It was the 20-year-old's first match since her second-round exit in January's Australian Open, and she had to respond to some early adversity after Kovinic broke her opening service game.
But the Brit answered right back, rattling off four games in a row to take control of the first set, and the second frame was nearly a carbon copy, falling down 2-0 before putting together five unanswered games of her own.
Meanwhile, world number 31 Danielle Collins – the highest ranked player in the draw to not receive a seed – was a surprise early elimination as Hungary's Dalma Galfi got the better of her 6-4 6-4. As a reward, Galfi will meet world number five Caroline Garcia in the second round.
Nuria Parrizas-Diaz set up an all-Spanish second-round clash with Paula Badosa after beating Italy's Elisabetta Cocciaretto 6-3 7-5, but fellow Spaniard Rebeka Masarova fell short 4-6 6-4 6-4 against American Peyton Stearns.
It was a great day for the Czech Republic representatives, with Katerina Siniakova earning a 7-5 6-4 over Julie Niemeier, Marketa Vondrousova dominating Rebecca Marino 6-2 6-2, and Karolina Muchova completing the hat-trick against Yulia Putintseva 6-3 4-6 6-4.
As good of a day as it was for the Czechs, it was just as poor for the Italians, with Lucia Bronzetti falling 7-5 4-6 6-3 to Bernarda Pera and Jasmine Paolini eliminated 7-5 6-1 at the hands of Tatjana Maria, joining Cocciaretto on an early flight home.
Number four seed Sabalenka was in fine form as she made a convincing start to her campaign, taking just over an hour to beat qualifier Perez 6-4 6-3 and move into the second round.
Perez was last year's doubles champion in Strasbourg but was no match for the world number 12, who is currently operating with her hitting partner as a coach.
Sabalenka had been working with Kindlmann, who used to coach her doubles partner Elise Mertens, heading into the US Open, but the Belarusian elected not to keep him on.
"I told him that I think I'm not ready for a new person on my team," Sabalenka explained.
"He's a great guy, great coach, but I'm just not ready. I'm going to work with my hitting partner for now as my coach."
Anna Blinkova, who beat Myrtille Georges 6-1 6-2, awaits Sabalenka in round two.
Former US Open champion Sloane Stephens succumbed to the shock result of the day, meanwhile, as she crashed out to world number 83 Nao Hibino.
Stephens was going up against an opponent who was 0-3 across her first three events since the season was restarted, but Hibino secured a hard-fought 6-2 3-6 6-1 triumph.
"Last week, I lost in qualies in Rome and it was not a good match, so I learned from that," Hibino said after the match. "Of course, Sloane is a good player, especially on clay courts. I prepared mentally and with tactics, so I think it helped a lot today."
It was a bad day for Stephens' compatriot Christina McHale, who went down 6-7 (4-7) 7-5 7-6 (7-3) to Zarina Diyas.
Magda Linette beat Pauline Parmentier to set up a tie with second seed Elina Svitolina, while third seed Kiki Bertens will meet Jelena Ostapenko.
There were also victories for Elena Rybakina, Ekaterina Alexandrova, Alize Cornet, Shuai Zhang and Katerina Siniakova.
Aryna Sabalenka began her Australian Open title defence with a convincing victory over Sloane Stephens at Rod Laver Arena.
Aryna Sabalenka recorded her 15th straight Australian Open victory on Sunday, teeing up a second-round meeting with Jessica Bouzas Maneiro.
The number two seed won 7-6 (10-8) 7-6 (7-3), saving seven set points after going 5-1 down in the first set, before roaring back to take both tie-breaks.
Sabalenka will face Marketa Vondrousova in the quarter-finals, with the Czech having produced a dominant performance in a 6-0 6-4 win against Kaia Kanepi.
"When it's 1-5 down, it's like you have nothing to lose, and you just go for your shots without thinking," Sabalenka said.
"That really helped me to stay in the set, to keep fighting and to keep trying."
Fourth seed Veronika Kudermetova eased through after a 6-4 6-0 victory against Bianca Andreescu, while Marta Kostyuk also advanced with a hard-fought 6-7 (5-7) 6-2 6-3 win over Elena Rybakina.
At the ASB Classic in Auckland, second seed Sloane Stephens is out after falling to Rebeka Masarova.
Having been forced to resume on Wednesday after rain suspended play the previous day, Stephens struggled against her Spanish opponent, eventually losing 6-3 7-6 (7-5).
Fifth seed Wang Xiyu is also out after retiring from her match against Karolina Muchova, but seventh seed Danka Kovinic is through after a simple 6-1 6-4 win against Nao Hibino.
Kovinic will face Lauren Davis next after she overcame Tamara Zidansek 6-2 6-1, while fellow American Sofia Kenin will take on number one seed Coco Gauff in the last 16 after she beat Wang Xinyu in straight sets.
After a hard-fought first set where both women were able to convert one of their four break point opportunities, Vekic took control in the tie-breaker and carried that momentum in the second set.
Vekic won 65 per cent of her service points in the second set, while Sakkari could only muster a 47-per-cent success rate, despite having a significant edge in first-serve accuracy (79 per cent to Vekic's 53 per cent).
The Croatian only had two break point opportunities, and that is all she needed, taking both to run away with one of the biggest wins of her career.
Vekic will play Karolina Pliskova in the second round after the Czech defeated America's Caroline Dolehide 6-3 6-4.
Pliskova, ranked 22nd in the world, had no issue with the world number 181 in one of Tuesday's biggest mismatches, nabbing a break-to-love in the first set before taking a quick double-break to begin the second.
World number 11 Daria Kasatkina was too strong for Leylah Fernandez, cruising to a 6-2 6-2 victory in exactly one hour.
Sloane Stephens earned a shot against world number five Aryna Sabalenka in the second round after getting the better of Switzerland's Jil Teichmann 6-1 7-6 (7-2).
World number 13 Garbine Muguruza was clearly not at 100 per cent, retiring hurt after losing the first five games of her match against China's Qinweng Zheng, and Coco Gauff will resume her contest against Robin Montgomery leading 6-3 3-2 after rain halted the action for the day.
The WTA Finals – which is contested by the women who finish top-eight for the season in ranking points – already had seven entrants locked in, with the eighth to be decided by the winner between world number six Sakkari and world number 12 Kudermetova.
Sakkari's ability to fight her way out of tough situations proved to be the difference as she saved 14 of the 15 break points she faced, while converting three of her nine chances. In the deciding third set, Sakkari made the most of her only break point, while saving all five of Kudermetova's chances to get back into the contest.
She will play Marie Bouzkova in the semi-final after the unseeded Czech was the beneficiary of Anna Kalinskaya's injury retirement in the first set, with Bouzkova leading 5-2.
It continues a terrific run of form for the 24-year-old Bouzkova, who is 19-3 in her past 22 matches, with her only losses in that span coming against world number 20 Karolina Pliskova, Wimbledon champion Elena Rybakina and world number two Ons Jabeur.
On the other side of the draw, Jessica Pegula breezed through the challenge of Sloane Stephens to win 6-2 6-2, which would have set up a tantalising all-American matchup against her doubles partner and rising star Coco Gauff, but she could not hold up her end of the bargain.
Gauff fell 7-6 (7-2) 4-6 6-3 to Victoria Azarenka, giving the Belarusian her third consecutive top-20 win after earning her spot in the quarter-final with victories over world number eight Paula Badosa and world number 15 Madison Keys.
Frenchwoman Tan, who is ranked outside the top 100, toppled the second seed in one hour and 46 minutes, winning 6-4 1-6 6-1.
In a seesawing battle, Tan's shotmaking and deft touch was a highlight, helping her steady after emphatically dropping the second set, before winning the first five games of the final set.
The win was Tan's first-ever victory against a top 30 player, qualifying her for the second round where she will play China's Wang Qiang who won 6-2 6-1 over Lauren Davis.
American sixth seed Sloane Stephens did not have such problems in her first-round encounter with qualifier Brenda Fruhvirtova, triumphing 6-2 6-2.
The 14-year-old Czech had enjoyed a good qualifying run but was thwarted by the 2017 US Open champion in 58 minutes.
Fruhvirtova had become the youngest player to compete in a WTA Tour main draw match since Keys in April 2009 at Ponte Vedra Beach.
Fifth seed Nuria Parrizas Diaz bowed out 6-1 6-4 to Slovakia's Ana Karolina Schmiedlova in one hour and 10 minutes. Schmiedlova will face Anastasia Potapova who progressed 6-4 2-2 after Lesia Tsurenko retired.
Japanese seventh seed Misaki Doi also exited, going down 4-6 7-5 6-4 to Marie Bouzkova in the late match, setting up a second-round date with qualifier Lucia Bronzetti who edged American wildcard Caty McNally 6-7 (5-7) 6-3 6-3.
Zheng Qinwen was another seeded casualty, with the Chinese eighth seed going down 6-3 2-6 6-2 to Anna Kalinskaya.
Williams had taken up a wildcard to play in the tournament following an early exit in Rome, where she was beaten by Nadia Podoroska in her opening contest in the clay-court swing.
Her campaign in Parma got off to a better start on Monday, a 6-3 6-2 result against WTA Tour debutant Lisa Pigato bringing a first triumph since losing in the semi-finals of the Australian Open in February.
However, the 23-time grand slam champion was unable to repel a determined Siniakova as she slipped out at the last-16 stage.
"It was a fantastic match and I played so well, I'm so happy that I could finish like that," Siniakova said after recording a 7-6 (7-4) 6-2 result.
"It was a pleasure to share the court with her.
"Now I will have the pressure because I defeated a great player, but I will enjoy it. I'm happy that I can continue and I will try to play my best in the next match."
Having failed to convert a set point opportunity, Williams hit back after losing serve with a break of her own to force a tie-break in the opener.
But Siniakova forced her way into a 5-3 lead before sealing it with a service winner, then quickly seized control of proceedings in a second set which was far less competitive.
Third seed Coco Gauff came through two tie-breaks to see off Kaia Kanepi, the American teenager having let slip a 5-1 lead in the second set before eventually sealing a 7-6 (8-6) 7-6 (9-7) win.
Petra Martic had no such issues in her opening contest against Varvara Gracheva, the second seed easing through 6-4 6-2 after one hour and 21 minutes on court.
Top seed Serena eased to a 6-3 6-2 triumph over teenage qualifier Lisa Pigato in the highlight of Monday's action in the clay-court tournament, wrapping up the result in 68 minutes.
Pigato, 17, was making her debut in the main draw of a professional event having come through qualifying in Parma.
She enjoyed early success against the legendary Williams, too, claiming a break of serve in the opening game, but the 23-time grand slam champion soon got to grips with the task at hand.
"The first game, she played really good and I needed to adjust to get back," Serena said afterwards. "It was a bit of both, figuring out her game as well."
Next up for the American, who accepted a wildcard into the event after an early exit at the Internazionali BNL d'Italia in Rome last week, will be Katerina Siniakova, a 6-1 6-3 winner over Clara Tauson.
Venus, however, failed to make it beyond her opening outing, losing out to Anna Karolina Schmiedlova despite taking the opening set.
The Slovakian made it four wins in five meetings against the seven-time major champion, eventually prevailing 5-7 6-2 6-2 after two hours and 39 minutes on court.
Elsewhere, Sloane Stephens defeated fellow American Caty McNally in two sets, while seventh seed Sara Sorribes Tormo overcame Bernarda Pera 6-4 6-2.
There was also a win in straight sets for eighth seed Caroline Garcia, who got the better of Paula Ormaechea in a contest that saw both players serve impressively.
