The 66th-ranked Sugita was no match for his illustrious opponent, although he did avoid a second-set bagel by breaking when Federer served for the match.
The 33-year-old Mayer squandered two break-points at 1-0 in the second set and paid the price as Federer broke again and never relinquished his advantage to win in just over an hour. The victory enabled him to reach the quarterfinal round in Halle for the seventh time.
Federer has yet to drop a set after three matches this week.
Federer started this tournament with his 1,100 match-win-he now trails only Jimmy Connors in that statistic-and he is rushing towards another remarkable milestone.More news: Kendall Jenner shares sexy bikini pic, underwear selfie
Gasquet will next play Germany's Alexander Zverev, last year's finalist who has won three titles this season, after he muscled past baseliner Roberto Bautista-Agut 6-7 (6) 7-6 (1) 6-1.
The All England Lawn Tennis Club (AELTC) use a unique seeding system that takes into account a player's Emirates ATP Ranking points as of 26 June in addition to 100% of the points earned for all grass-court tournaments in the previous 12 months. Big and strong, and super excited to be on the tour, and working hard. "It could be somewhat similar to how I played today".
"I don't know him that well yet so it's a match where I'll most likely focus on my own game", Federer said of facing the Russian in the semifinals.