The top seed, who has yet to drop a set in the tournament, beat Mayer 6-3 6-4 with his ninth Wimbledon title firmly in his sights.
"After I got that in the bag I was really able to start to relax and really play and feel the way I want to feel out there".
As for the defending champion, Mayer has not had the best of seasons so far. His problem was that he played, in Federer, an eight-time champion who, after a rusty return to grass in Stuttgart, was finding his groove on the turf. He faced Mischa Zverev in the next round and the German also proved to be no match for him as he finished him off in straight sets.
The 19-year-old Rublev, who received a wild card for the grass-court tournament, will face another Russian in Karen Khachanov in the next round.
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Federer overcame an uncharacteristic wobble by saving two break points in his first service game of the second set and broke three games later, when a double fault followed by a wide backhand saw Mayer fall behind again.
It will be their eighth career encounter and fourth in Halle, with Federer dominating the head-to-head series 7-0. I don't know him that well yet so it's a match where I'll most likely focus on my own game.
Such invention only earned the 33-year-old a stay of execution, though, Federer holding to seal it soon afterwards with a trademark off forehand victor into empty court.
The 18-time Grand Slam victor, seeking to move clear of fellow tennis legend Pete Sampras next month by claiming a record eighth singles title at SW19, claimed a 7-6 6-4 straight-sets victory over the spirited Zverev to book a Halle quarter-final spot and crucially ensure top-four seeding for the third major of the year.