Second-seeded Federer had little trouble in seeing off Argentina's world number 34 Leonardo Mayer 6-1, 6-2, 6-2 in just 77 minutes on Arthur Ashe Stadium.
The 34-year-old Swiss, bidding to become the oldest champion in New York in 45 years, fired 12 aces, 29 winners and broke serve six times.
"I feel good now. I actually wasn't so confident yesterday. I just felt like maybe it could be one of those matches I just couldn't see coming," said Federer.
It was the 10th retirement in the men's event in the first round; there have also been two pullouts in the women's draw.
Third seed Murray downed the sport's latest bogeyman, Nick Kyrgios, 7-5, 6-3, 4-6, 6-1 for his fourth win in four meetings against the Australian and third in three Grand Slam matchups this year.
The British star, who has made at least the quarter-finals in New York every year since 2010, goes on to tackle France's Adrian Mannarino.
"It was tough, very humid, I had to do a lot of running and defending," said Murray after his night match.
Kyrgios came into the tournament under the shadow of a suspension after his now infamous and lurid Montreal rant at Stan Wawrinka, but that threat only applies to ATP events.
So it was no surprise that the first profanity from the Australian came after just four games of the first set, when he complained about fans taking their seats during points.
His Davis Cup teammate, Thanasi Kokkinakis, was also knocked out, retiring on one leg in the fifth set against French 12th seed Richard Gasquet.
The 19-year-old Kokkinakis, who was caught up in Kyrgios's infamous slur against Wawrinka, suffered severe cramping.
But he battled on, serving underhand and forced to play at walking pace before he finally gave up, losing 4-6, 6-1, 4-6, 6-3, 2-0 after almost three hours in 30-degree heat.
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