"Two matches in a row, it's beautiful right now," Federer, who pulled out of the French Open with a back injury, said of his grasscourt buildup ahead of this month's Wimbledon.
The 34-year-old seven-times Wimbledon champion will now play rising Austrian Dominic Thiem in the semi-finals.
Federer looked to be moving well on the slick grass and his serve clicked throughout apart from when he was broken by Mayer early in the second set.
He battled back from 15-40 in the 12th game of the second set and then saved another set point with an unreturnable serve.
Federer was immaculate in the tiebreak, treating the crowd to some glorious winners as he strolled into a 6-0 lead.
He claimed victory with a clean winner on his second match point -- moving ahead of Ivan Lendl into second place on the all-time list for ATP career match wins with 1,072.
(Reporting by Martyn Herman; Editing by Toby Davis)
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