World number two Federer was barely ruffled as he cruised to a 6-3, 6-2 win over Nick Kyrgios at Chicago's United Center on Saturday.
Earlier, Zverev saved a match point to defeat John Isner in day two's opening singles rubber.
Zverev, 21, dug deep to hold off the big-serving Isner to win 3-6, 7-6 (8/6), 10-7.
Under the format being used for the tournament, a 10-point tiebreak is used instead of a decisive third set.
After dominating Friday's opening day, winning three out of the four matches, Europe's progress looked to be checked after Isner grabbed the first set against Zverev.
But rising German star Zverev showed great determination in saving a match point at 6-5 down in the second-set tie-break to level the contest.
Zverev then raced into the lead in the 10-point match tie-break before closing out for the win.
Under the scoring system for the Laver Cup, victories on day two are worth two points each, with victories on day three rising to three points.
Europe needs only 13 points to take the title, which is being held for the second time.
In later matches on Saturday, Novak Djokovic faces Kevin Anderson, while Europe doubles pairing Grigor Dimitrov and David Goffin complete the schedule against the World team's Jack Sock and Kyrgios.