The Swiss second seed was untroubled on his way to a 6-4 6-3 6-2 victory over Spanish 17th seed Bautista Agut on Arthur Ashe Stadium.
Federer, targeting a 17th career grand slam title, was at his consummate best, producing 36 winners to his opponent's 20 as he booked his spot in the last eight.
Awaiting Federer is 20th-seeded Frenchman Gael Monfils.
Federer needed one hour 56 minutes to see off Bautista Agut, who was into the fourth round of a major for just the second time in his career.
The Swiss, meanwhile, was in his 49th fourth-round battle at a grand slam - the level he was ousted at in 2013 at Flushing Meadows - but history would not repeat.
Federer coasted to a double-break 5-1 lead in the first set, but Bautista Agut did not throw in the towel at the first time of asking, the Spaniard breaking back to close to 5-4.
The 33-year-old managed to hold serve at the second time of asking to wrap up the set, and the opener would prove to be Bautista Agut's best chance of taking a set.
Breaks in the third and ninth games of the second set saw Federer double his advantage, and the five-time champion - despite facing seven break points for the match - was only broken once as he wrapped up proceedings in relatively quick time.