Federer, who has an Olympic gold medal in doubles, and Zverev won 6-3, 7-5 against Jack Sock and Denis Shapovalov as the Europeans swept the two-match evening session. The afternoon singles matches were split at the noisy, 17,000-seat Palexpo arena.
"Thanks Sascha, for carrying me in that second set," Federer told his playing partner in an on-court interview.
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. Shapovalov let three match points slip in the decisive third set, which is played as a super-tiebreaker.
Sock, whose rankings fell into the 200s during an injury-hit season, then leveled the three-day event by winning 6-1, 7-6 (3) against No. 11 Fabio Fognini.
Federer is back on court before another sold-out crowd Saturday, when he will play Nick Kyrgios in the afternoon's second match.
"I'm fresh, I'm ready to go for more," said Federer, who co-owns the Laver Cup competition.
Saturday's play opens with Zverev playing John Isner. Rafael Nadal enters the stage for Europe in an evening singles match against Milos Raonic, then stays on to partner with Tsitsipas against Kyrgios and Sock.
Victories on Saturday in the three singles matches and one doubles are worth two points each. Sunday's matches score three points for the winning team, and 13 points is needed to lift the title.
Europe seeks a third consecutive victory in the Laver Cup. It was launched two years ago in Prague, then moved to Chicago last year.