The 17-time grand slam champion struggled early before winning 4-6 6-1 6-1 6-1 on Arthur Ashe Stadium.
Federer won 20 of the final 24 games of the third-round match in New York to make it three wins in as many meetings with Granollers.
The Swiss second seed had battled in the humidity during the first set as Granollers took a 5-2 lead.
But a two-hour rain delay, which led to Federer entering a largely empty stadium, helped the five-time US Open winner.
Federer took complete control after dropping the first set.
He hit 17 winners in the second set and another 17 in the third, breaking serve five times in that time.
Federer broke another three times in the fourth set as he completed what proved to be a comfortable win.
He finished with 57 winners and 27 unforced errors and will meet Spanish 17th seed Roberto Bautista Agut.