Reigning US Open champion Alcaraz eased past Bagnis 6-0 6-2, firing 12 winners and three aces for the match, committing only 11 unforced errors.
The Spaniard won the first set in 24 minutes, marking the first time he has won an opening frame 6-0 at an ATP Masters 100 event. It was also the first bagel he has dished out since beating Bagnis in Umag last year.
"I knew that I needed to be focused for my first match," Alcaraz said after his win, having triumphed in Indian Wells last week. "To start a new tournament is never easy, different conditions.
"I have to be ready to get used to these new conditions, but I was really focused on the match from the beginning, and I'm happy with the way that I played. I try to improve every day."
Alcaraz will face Dusan Lajovic after he beat 30th seed Maxime Cressy 6-4 7-6 (7-2), having knocked off Andy Murray in the first round.
Zverev was the highest seed to bow out on Friday, going down 6-0 6-4 to Japan's Taro Daniel, who beat world number four Casper Ruud last month in Acapulco.
Daniel triumphed in 73 minutes against the German, hitting 17 winners and committing only two unforced errors for the match, while Zverev gave up 18.
The Japanese wild card will meet Emil Ruusuvuori in the third round after the Finn beat 22nd seed Roberto Bautista Agut 6-4 7-6 (7-5) to open up that section of the draw.
Third seed Ruud also had few problems getting past Ilya Ivashka 6-2 6-3 to set up a clash with Botic van de Zandschulp after his three-set win over Alexei Popyrin.
Sixth seed Andrey Rublev overcame J.J. Wolf 7-6 (7-3) 6-4, having been broken in the opening game of what was a tight contest.
Jannik Sinner beat Laslo Djere 6-4 6-2, marking his 16th win out of 20 appearances at the Sunshine Double (Indian Wells and Miami), with only three other players – Novak Djokovic, Alcaraz and Hubert Hurkacz – having as many wins within their first 20 matches.
Ninth seed Taylor Fritz defeated fellow American Emilio Nava 6-4 6-1, and will face Denis Shapovalov in round three after he beat Guido Pella 6-3 3-6 6-3.
American 16th seed Tommy Paul came from a set down to beat Marc-Andrea Huesler 5-7 6-3 6-4, while seventh seed Holger Rune beat Martin Fucsovics 6-3 7-5.
Murray, a five-time finalist in Melbourne, defeated Nikoloz Basilashvili 6-1 3-6 6-4 6-7 (5-7) 6-4 in his first-round match on Tuesday, yet the Scot was no match for Japan's Daniel two days later.
Perhaps fatigue played its part, with Murray struggling to find his rhythm against the qualifier, who is ranked 120th in the world.
Murray had never before lost to a player ranked as low as Daniel, who has reached the third round of a grand slam for the first time.
Daniel sealed a 6-4 6-4 6-4 success with a neat backhand volley, taking the match at the first time of asking.
Murray, who had been targetting a "deep run" in the tournament, walked off court in disgruntled fashion, offering a quick acknowledgement to the crowd after his first appearance in the season's opening major since 2019.
Asked in his post-match news conference if he planned to return to Melbourne next year, Murray told reporters: "Yes, but not if I do what I did tonight too often.
"I want to perform well on the big events and what I did tonight was not good enough. Reaching the second round of grand slams doesn't particularly motivate me."
While Murray was far from the level he displayed against Basilashvili, Daniel's performance was more than worthy of victory.
"Winning a big match like this is unbelievable," the 28-year-old said. "It was an amazing level from me, I was getting nervous in the third set.
"I tried not to make a big deal about this – everyone said I was playing Murray – but I tried to treat it like another match."
Murray, meanwhile, did not use fatigue as an excuse.
"I felt alright physically so I was pleased from that perspective," he said. "I just made way too many errors on the court."
The statistics back up Murray's claim. He won just two out of 11 break points and made 49 unforced errors, with Daniel only making 21.
Daniel struck 12 aces to Murray's seven, recording a first-serve win percentage of 79.
Murray's first-serve win percentage was down at 66, while he also made three double faults to Daniel's two.
America's Nakashima – who took Nick Kyrgios to five sets at Wimbledon – was the highest seed in action as things kicked off on Monday, and the difference in the match proved to be his quality ground strokes.
Both players won at least 79 per cent of their successful first serves, but while Thompson's effectiveness dipped to 39 per cent on his second serve, Nakashima's stayed up at 65 per cent, only allowing two break point opportunities all match.
Nakashima will meet John Millman in the next round after Millman defeated fellow Australian Alexei Popyrin 7-6 (7-1) 4-6 6-3.
Popyrin clearly had the serving advantage, winning the ace count 12-to-four, but in a match where each player only one had break of serve each, Millman found the important points when it really mattered.
Japan's Taro Daniel came back from a rough start to handle the challenge of America's Sebastian Korda 1-6 6-1 6-3.
Korda was in total control early, winning 12 of his 13 points on serve (92 per cent) in the opening set, but that number dipped sharply to 37 per cent (13-of-35) as Daniel turned the match on its head and found a way to counterpunch the American.
The last singles contest saw Kwon Soon-woo win a hard-fought clash with Marcos Giron 7-6 (7-3) 4-6 7-5. The match was so closely contested that Kwon won 51 per cent of the total points (104-of-204), while Giron won 49 per cent (100-of-204).
Meanwhile, Kyrgios was the biggest name in action, taking the doubles court in his Australian Open-winning partnership with Thanasi Kokkinakis, defeating Nicolas Mahut and Edouard Roger-Vasselin 7-6 (10-8) 6-3.
Karatsev showed astonishing defiance to beat the world number one 7-5 4-6 6-4 in the longest ATP Tour match of the year on Saturday.
The third seed from Russia saved 23 of the 28 break points he faced as his aggressive approach paid off, toppling the 18-time grand slam champion in a contest that lasted three hours and 25 minutes.
It was sweet revenge for Karatsev, who was beaten by the legendary Serbian at the semi-final stage of the Australian Open two months ago.
Djokovic had won 11 matches in a row in his homeland, but bowed out despite being 2-0 up in the first two sets as a solitary break in the decider ended his run.
Karatsev, the world number 28, said: "It was a long, tough match [against a] tough opponent.
"You have to put [in] like 200 per cent to beat this guy, it's like playing against a wall. And he also made some good shots.
"He doesn't give you any free points. He always makes you play and you have to be always there because once you miss a couple of shots, he just takes it very quickly. That’s how I lost the second set."
Berrettini secured his place in the final with a 6-1 6-7 (5-7) 6-0 victory over lucky loser Taro Daniel.
Second seed Berrettini only lost six points in the last set after Daniel came from a break down in the second before winning a tie-break to take the second semi-final the distance.
Heading into Sunday's final, Karatsev and Berrettini have never previously faced each other on the ATP Tour.
Murray, 34, began slowly but ground out a 1-6 6-2 6-4 win over a player he has faced three times in the early stages of this season.
The former rankings leader has bounced back into the ATP top 100 after career-threatening injury woes and is looking to push on from number 88, his current position.
He will play 31st seed Alexander Bublik next.
Murray was delighted to reach the 700-win mark, telling Amazon Prime: "It means a lot. It's a target I set myself towards the end of last year. With everything that's gone on in the last few years, it's not been easy to get there.
"It's a great achievement; not loads of guys have been able to do that, so I'm really happy about it."
The three-time grand slam winner becomes the fourth active men's player to reach 700 wins, after Roger Federer, Rafael Nadal and Novak Djokovic.
Murray was beaten by Daniel in the second round of the Australian Open before avenging that in the Qatar Open.
The Scot has 14 titles to his name from Masters 1000 events but surprisingly has never triumphed at Indian Wells, and he was given a thorough workout by qualifier Daniel.
Daniel dashed into a 5-1 lead, striking the ball more cleanly and moving better than Murray, who has recently agreed to bring Ivan Lendl on to his team for a third time.
The Japanese star had no trouble seeing out that opening set, but the second was dominated by Murray.
Daniel then won the opening eight points of the decider to surge 2-0 ahead, but Murray broke back in the sixth game. Murray saved break point at 4-4 and made Daniel serve to stay in the contest. That proved beyond the 29-year-old, with Murray carving out three break points and taking the third of those when Daniel sent a forehand long.
Murray said he took pride from the match because he "figured it out" after being outplayed early on.
He said Lendl had wished him luck before going on court, with the pair to begin working together after Murray plays the next event in Miami.
The fifth seed found a crucial break of serve when leading the opener 5-4 to take the first set in Kazakhstan, before winning the next three games to put himself in the ascendancy in the second.
And despite a desperate last stand to stay in the match from Djere, Rublev held serve three times with relative ease to finish off his Serbian opponent.
Eighth seed Felix Auger-Aliassime served 14 aces to Roberto Bautista-Agut's two in their first-round match, but it was the Spaniard who advanced with a 6-4 7-6 (8-6) success.
Meanwhile, world number 10 Hubert Hurkacz overcame Francisco Cerundolo in straight sets.
Adrian Mannarino, the runner-up in Astana in 2020, came from a set down to beat Stan Wawrinka – playing as a wildcard – 1-6 6-2 6-3, and qualifier Zhizhen Zhang romped to victory with a superb third set to seal a 4-6 6-4 6-1 comeback win of his own over Aslan Karatsev.
At the Tokyo Open, fourth seed Frances Tiafoe cruised into the second round with a 6-3 6-4 triumph over Yasutaka Uchiyama, one of three Japanese players who crashed out at the first-round stage of their home tournament.
Uchiyama's countrymen Taro Daniel and Shintaro Mochizuki also tasted defeat at the hands of Bernabe Zapata Miralles and Brandon Nakashima, respectively.
The top seed was given a second set scare by the Chilean but wrestled back control to emerge as the victor in a 6-2 3-6 6-3 encounter.
The two-time grand slam finalist, having been runner-up at Wimbledon and the US Open this year, will face Yoshihito Nishioka after he saw off Japanese compatriot Taro Daniel in a 6-2 6-4 win.
His victory also ensured his place at the ATP Tour's end-of-year finale in Turin, alongside Spanish duo Rafael Nadal and Carlos Alcaraz.
Cameron Norrie remains on course to set up a potential final in Korea with Ruud after brushing aside Kaichi Uchida with a comfortable 6-2 6-2 victory.
The Briton will face eighth seed Jenson Brooksby after he turned over home favourite Kwon Soon-woo in a 6-3 6-4 result in Seoul.
Brooksby's fellow Americans Mackenzie McDonald and Aleksandar Kovacevic also came through their last-16 encounters to set up a quarter-final clash against each other.
The 2021 Wimbledon semi-finalist needed 109 minutes to claim a 7-5 6-4 win over Munar, assuming control by breaking in the seventh game of the opener before doing likewise early in the second set.
Canadian Shapovalov – who is seeded fourth for the event – had not featured since a dramatic loss to Andrey Rublev at the US Open earlier this month, but produced a consistent display in Seoul, not committing a single double fault during the match.
His reward for the victory is a last-eight tie with Radu Albot, who held his nerve in two tie-breaks to record a 7-6 (7-3) 7-6 (7-3) win over Steve Johnson.
In Wednesday's other match, Japan's Taro Daniel beat Ecuador's Emilio Gomez in straight sets in the duo's round-of-32 tie.
Top seed Casper Ruud begins his campaign against Nicolas Jarry on Thursday.
Daniel started beautifully, securing a break in the first game of the match, and his superior returning ability would carry him through the first set, winning 51 per cent (21-of-41) of his return points in the opening frame.
It was Daniel's serve which controlled the second set, winning 80 per cent (16-of-20) of his service points while allowing no break point opportunities for Nava, finishing the job in 94 minutes.
He will play world number 25 Evans next, who he has beaten in two of their past three meetings, although they have only gone head-to-head once since 2015 – in 2019, when Daniel won in straight sets.
Meanwhile, Mitchell Krueger ended a three-match losing streak against American compatriot Christopher Eubanks with a 6-4 6-4 triumph.
Eubanks had the power advantage, winning the ace count nine-to-four and the winner count 25-to-18, but he was also far sloppier, committing all four of the match's double faults as well as totalling 20 unforced errors to Kreuger's four.
In a clash between two veterans, 30-year-old Denis Kudla proved to be too much for 38-year-old Fernando Versaco, winning 6-3 6-3 to relegate the Spaniard to four losses from his past five matches.
In the final match of the night, Brandon Nakashima overcame Zachary Svajda 7-5 7-6 (7-5) after a gruelling 69-minute second set, contesting 96 points in the second frame after 66 in the first.
