Five-time champion and world number two Federer claimed a 6-3, 6-4, 6-4 victory over his 29th-ranked German opponent.
The 34-year-old Swiss star, hoping to become the oldest champion since Ken Rosewall in 1970, will next face US 13th seed John Isner, who progressed after Czech opponent Jiri Vesely retired with a neck injury after dropping the first two sets.
Bouchard, meanwhile, was facing a battle to make her scheduled last-16 clash with Italy's Roberto Vinci on Sunday after injuring her head in a freak fall in the women's locker room on Friday.
"The severity of the injury was and continues to be assessed by the on-site medical personnel at the US Open," said tournament director David Brewer.
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"No determination about competing in the singles competition has been made at this time. Bouchard will continue with medical evaluation and treatment for the remainder of the day.
"An update on her condition will be provided later today."
Federer has now reached the fourth round in New York for the 15th year as he looks to return to the final for the first time since 2009.
"I won the big points today, I served well when I had to, even though it was up and down for both of us," said Federer, who will take a 4-1 career lead over Isner into their fourth-round clash.
"He stood far back to return so I had to do a reset in my mind."
Two-time runner-up Victoria Azarenka, the 20th seed, triumphed over 11th-seeded German Angelique Kerber in an Arthur Ashe Stadium thriller lasting almost three hours.