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.
The 17-time Grand Slam champion goes on to face Belgium's Steve Darcis, who was handed a place in the next stage when Cypriot veteran Marcos Baghdatis retired injured with a groin strain.
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.

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