Federer, seeking his sixth New York title, bested American Donald Young 6-3, 6-2, 6-4.
The Swiss superstar, coming off his seventh Wimbledon crown, is tied with Jimmy Connor and Pete Sampras with five U.S. Open victories.
Britain's Murray, meanwhile, rolled past 73rd-ranked Russian Alex Bogomolov Jr. 6-2, 6-4, 6-1 in a match played under humid and breezy conditions at Arthur Ashe Stadium.
The Olympic gold medalist now goes on to face Croatia's Ivan Dodig, who advanced with a 6-0, 6-1, 6-2 pasting of Japan's Hiroki Moriya.
Also posting first-round wins were Japanese Kei Nishikori, American Jack Sock and 24th-seeded Spaniard Marcel Granollers, who defeated American Denis Kudla 6-3, 4-6, 6-3, 7-6 (7-2).