Bayern Munich will face Real Madrid in the first leg of the Champions League semifinal on Tuesday
Zverev fired a cross-court forehand winner to clinch a 6-4, 3-6, 10-4 victory in the decisive singles match and give the Europeans a 13-11 win in points.
After Zverev fell to the ground in triumph, he disappeared from view under a pile of celebrating teammates including Federer, Rafael Nadal and their captain Bjorn Borg.
Tennis great Rod Laver, for whom the event co-owned by Federer is named, presented the trophy to the Europeans, watched by the world team captained by John McEnroe.
"I played an unbelievable tiebreaker," Zverev said. "I'm super happy and super thankful to Roger and Rafa and the rest of the team. Without them on the bench today, I couldn't have done it."
Two hours earlier, Federer had delighted a sold-out crowd of 17,000 in his home country by beating John Isner 6-4, 7-6 (3) to set up the winner-take-all finale.
The European team also won the first two Laver Cup editions, played in 2017 in Prague and last year in Chicago.
Europe began Sunday leading 7-5 in a scoring system that gave one point for a win on Friday and two on Saturday. Victories in the doubles and each of three singles matches scored three points on Sunday.
Nadal scratched from Europe's intended lineup in both the opening doubles with Federer and the first singles due to an inflamed hand.
Isner and doubles specialist Jack Sock then beat Federer and Stefanos Tsitsipas 5-7, 6-4, 10-8.
Nadal's replacement in singles, No. 5 ranked Dominic Thiem, was beaten 7-5, 6-7 (3), 10-5 by Taylor Fritz in a second straight match decided by the super-tiebreaker.
Fritz was a late replacement for Nick Kyrgios, who like Nadal scratched after both had played in singles and doubles on Saturday.
Strong play by the Americans on Sunday gave the world team an overall lead for the first time in the three-day competition.
Isner and Sock improved to 3-0 in Laver Cup doubles, and the 30th-ranked Fritz overcame Austria's Thiem.
"This has to be one of the biggest wins of my career," Fritz said. "I'm the kind of person that really doesn't want to let my team down."
With Europe needing to sweep the last two rubbers, Federer came through in a typically tight match against Isner with few break-point chances.
"I'm thrilled that I was able to give something back to the team," Federer said. "Obviously to see Rafa on the sidelines after having to pull out with an injury, it's great camaraderie I feel."
Results
Singles
Roger Federer (Europe) bt John Isner (World) 6-4, 7-6 (7/3)
Taylor Fritz (World) bt Dominic Thiem (Europe) 7-5, 6-7 (3/7), 10-5
Alexander Zverev (Europe) bt Milos Raonic (World) 6-4, 3-6, 10-4
Doubles
John Isner and Jack Sock (World) bt Roger Federer and Stefanos Tsitsipas (Europe) 5-7, 6-4, 10-8
Final overall score: Europe 13 World 11
Note: On opening day, each win secures one point, on second day, two points, on third and final day, three points
Bayern Munich will face Real Madrid in the first leg of the Champions League semifinal on Tuesday
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