The European pair saw off Shapovalov and Sock 6-3, 7-5 after a dramatic second set.
"All of us tennis players travel around the world for 10, 11 months of the year," Federer told a capacity 18,400-crowd at the Palexpo arena.
"There's no place like playing at home, so this is a very special night for me."
How did the matches unfold?
American Sock had beaten Italy's Fabio Fognini 6-1, 7-6 (7/3) in the singles to draw John McEnroe's visitors level.
Prior to that, Dominic Thiem has registered a win against Shapovalov.
With the proceedings being 1-1, Greek star Stefanos Tsitsipas restored Europe's advantage by overcoming Taylor Fritz in a super tie-break to win 6-2, 1-6, 10-7.
Federer and Zverev then increased the lead to 3-1.
Zverev says how Federer and Nadal inspired him
"I had two great coaches, one on the court and one of the sideline (Nadal) telling me what to do every point," said German Zverev. "It was perfect. I could shut down my brain a little bit and do what they told me."
Team Europe favorites to clinch the Laver Cup
Holders Europe are the favorites to defend the title this year.
Last time, Team Europe won the 2018 edition 13-8 in Chicago.
Meanwhile, the scoring system sees one point awarded for each win on the first day (Friday), two points allotted for each victory on the second day, and three points being offered in each match on Sunday's final day.