Team Europe captain Bjorn Borg chose to rope in Stefanos Tsitsipas as Nadal's replacement for the doubles match with Roger Federer, while Austria's Dominic Thiem will lock horns with Australia's Nick Kyrgios in the men's singles match.
But after Federer beat John Isner, Germany's Zverev kept his cool in an electrifying atmosphere to beat Raonic 6-4 3-6 10-4.
Zverev fired a cross-court forehand victor 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.
They have won all three editions of the Laver Cup since it was conceived in 2017 - Zverev has factored into each of those victories.
Shapovalov let three match points slip in the decisive third set, which is played as a super-tiebreaker. "This event is something I hope to play in every single year of my career". "I'm super happy and super thankful to Roger and Rafa and the rest of the team".
Nadal got the favor back from Federer during a 6-3, 7-6 (1) win over Milos Raonic that opened the evening session.
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"I don't know what to tell you other than we wanted him to come out 100% and he didn't feel like he could".
It somehow didn't work, as Fognini would drop the match, giving Sock his first singles win of 2019 and Team World's first-ever Day 1 singles victory. "And how they were on the bench and how everybody was on the bench was an unbelievable feeling for me to win in the end".
It meant that if Federer lost to Isner, Team World would win the title.
The six-man Europe team has a big advantage in ATP rankings, and No. 5 Dominic Thiem started the day by beating the 33rd-ranked Shapovalov 6-4, 5-7, 13-11.
Alexander Zverev fought the nerves and got over a tough season by winning the Laver Cup for Team Europe.
The Aussie unhealthy boy, competing for Staff World, complained to captain John McEnroe that he misplaced his focus early within the second set after a girl within the stands caught his eye. It's been a dream weekend for me to be playing in Switzerland.