The number one seed lost the opening set but responded to take the following two and win 3-6 6-2 7-5.
Alcaraz dominated at the net, winning 30 of 39 points to edge out his opponent after more than two hours.
Elsewhere, at the Vienna Open, fourth seed Taylor Fritz overcame Yoshihito Nishioka in another three-set thriller.
Nishioka took the opener, and Fritz was staring down the barrel in a second-set tie-break before trailing by a break in the decider.
However, the American won the final five games to advance 6-7 (4-7) 7-6 (9-7) 6-3.
The comeback boosted Fritz's chances of making the ATP Finals, trailing Felix Auger-Aliassime by 315 points in the Race to Turin.
Third seed Andrey Rublev found life far easier as he beat Diego Schwartzman 6-4 6-1.
The Canadian had not won any of his eight career finals before this season, but he has now come through four of five in 2022.
Three of those have been in Auger-Aliassime's past three tournament entries, as he followed up successes in Florence and Antwerp by beating Holger Rune on Sunday.
Rune had himself won last time out against Stefanos Tsitsipas in the Stockholm final, but he was brushed aside by Auger-Aliassime 6-3 7-5.
The world number nine threatened to wrap up the victory in double quick time as he raced through the opener, with Rune responding in the second set to drag the match out to 100 minutes.
Auger-Aliassime fended off a break point early in the second and two more later in the set, protecting his serve throughout and then going on the offensive.
The fourth of four break opportunities at 5-5 was seized, and Auger-Aliassime – fresh from beating world number one Carlos Alcaraz – served out the match at the third attempt.
The 22-year-old has never previously made the season-ending Finals but moved ahead of Andrey Rublev with his latest success.
Nadal suffered from a recurring foot problem last season but returned to secure a record 21st grand slam title at the Australian Open in January, moving ahead of Federer and Novak Djokovic in the men's all-time list.
Federer has been out of action since losing in the Wimbledon quarter-finals last year, where he sustained another problem with his knee and subsequently underwent a third surgery in the space of 18 months.
The Swiss is yet to put a timeframe on his full ATP Tour return, although he is scheduled to play at the Laver Cup in September before playing the Swiss Indoors Basel event in his home city in October.
Federer, who turns 41 in August, referenced Nadal as he expressed his hopes to emulate the Spaniard's 20-match winning streak that he embarked on when returning from injury this season.
"It's incredibly inspiring when someone comes back from massive health problems," Federer told Caminada Magazin.
"Rafa and I talk on the phone from time to time, we talk a lot. I knew he wasn't doing great, but when he made it I was really happy for him. The effort is immense."
As for Federer's recovery, the world number 46 detailed the struggles he has to go through just to make it onto the court.
"As with a car, you have to turn a thousand screws until the engine runs smoothly," he added. "Today, mobilisation, stretching, and a warm-up in the morning take about 45 minutes. Then we drive to the plant. There follows a warm-up on the pitch, half an hour.
"After that I eat, stretch, strengthen my ankles with tapes, then warm up again, do gymnastics and explosive speed exercises. Before I finally play, I took care of my body for two and a half hours.
"I don't post many pictures of the strenuous training because I was always convinced that it was a matter of course. Everyone trains hard.
"I swore to myself that by the end of my career I wouldn't be completely broken. Later I would like to go skiing with the children and play football with my colleagues. That's why I'm doing rehab now – not just for tennis. Also for life after your career."
Asked when he will make his comeback, Federer added: "I can't even think that far. I'm waiting for the doctors' okay. I'm ready to give it my all again.
"I feel like a racehorse scratching its stall and wanting to race. In the summer I hope to be able to hit the ground running.
"I'm looking forward to coming home in the evening after the tough day of training and being completely exhausted."
It means the legendary 20-time grand slam winner is set to miss all four majors this year.
The eight-time Wimbledon champion will turn 41 in August, and it remains to be seen whether the two events turn out to be farewell tournaments before he retires.
Federer has battled knee trouble over the past two years and has barely featured on the ATP Tour, with his most recent appearance coming at Wimbledon last year, where he was beaten in the quarter-finals by Hubert Hurkacz, a straight-sets defeat that culminated in a 6-0 third set.
The Swiss great has undergone surgery in an effort to finish his career on his own terms.
News of his comeback plans were announced by the Swiss Indoors organisers, who issued a statement that said: "Ten-time singles champion and hometown hero Roger Federer has announced his comeback to the stadium at St Jakobshalle. The Swiss all-time great has confirmed his initial agenda will include the Laver Cup in London followed by the Swiss Indoors in Basel."
Responding to the announcement, Federer posted on Instagram: "Looking forward to playing back home."
The Laver Cup takes place in London from September 23 to 25, and the Swiss Indoors Basel runs from October 24 to 30.
Federer will play his opening match at the latter on Tuesday, October 25, the organisers said.
"The worldwide interest in the return of the hometown hero and 20-time grand slam champion to the ATP Tour is expected to be tremendous," the Swiss Indoors statement added.
Federer for a long time held the record for the most grand slam singles titles won by a man; however, he has been joined on 20 by Novak Djokovic, and Rafael Nadal moved out on his own with 21 by triumphing at the Australian Open this year.
Former world number one Murray pulled off a remarkable win over Russian Roman Safiullin, winning five games in a row from 4-1 behind in the final set.
But that was just a tease for the excitement that followed, as Wawrinka, who like Murray won three grand slam titles in his prime, felled second seed Ruud in straight sets.
The 37-year-old Swiss delighted his home crowd with a 6-4 6-4 victory over this season's French Open and US Open runner-up, serving nine aces as he defied a world ranking of 194th to produce some of his old magic.
Murray has fallen away from the elite ,too, but at 49th in the world he is not far away from rejoining that pack, although defeat looked to be beckoning against Safiullin.
The 35-year-old Scot fist-pumped with relief at recovering the break of serve that Safiullin snatched early in the decider, and Murray hit a service winner on his second match point to seal a 6-7 (5-7) 6-3 6-4 success.
The victory for two-time Wimbledon winner Murray took him to 26 wins this season, the most he has had in a single campaign since pulling off an astounding 78 victories in 2016, the year he won at the All England Club for a second time, landed a second Olympic gold medal and finished top of the rankings.
Murray said of his win: "As the match went on, I started to dictate a few more points and served better."
Quoted on the ATP website, he added: "I changed the way I was returning a little bit, and when I did that, I was able to create a few more opportunities and frustrate him a little bit."
Pablo Carreno Busta scored a 6-2 6-1 win against Argentinian Sebastian Baez, with Alex Molcan, Dominic Stricker and Botic van de Zandschulp also advancing to the last-16 stage.
At the Vienna Open, Murray's fellow British players Cameron Norrie and Dan Evans both went through to the second round, Norrie beating Argentina's Pedro Cachin 3-6 6-2 7-6 (7-1) and Evans fending off Germany's Oscar Otte 6-4 7-6 (7-3).
Canadian left-hander Denis Shapovalov beat Austrian wildcard Jurij Rodionov, but the home crowd had something to celebrate when former US Open champion Dominic Thiem continued his resurgence by eking out a 2-6 7-6 (7-2) 7-6 (8-6) victory against American Tommy Paul, saving two match points.
