Williams missed last year's Wimbledon as she was pregnant with her first child.
Two successive aces, clocked at 103 and 106 miles per hour, gave the 21-year-old the first set and the 2017 French Open champion tightened her grip on the contest when she ended another flurry of breaks to go 4-2 up in the second.
Giorgi appeared a bit salty - maybe that's her normal demeanor - as she shook hands with Williams at the net after the match and quickly grabbed her bag and walked off the court.
That was winning majors, and Williams enters the semifinals as the lowest-seeded player (25th) remaining in this year's Wimbledon tournament.
"After the first set, I said "okay let's go three sets", I just kept fighting".
Williams had made a tentative return to action following complications during daughter Olympia's birth, struggling in her few appearances before pulling out midway through the recent French Open with a pectoral injury.
Serena Williams has survived another day of Wimbledon carnage to edge ever closer to a record-equalling 24th grand slam singles crown. Even you're returning on the line, under her legs, she can go for a victor, it's insane. "Sometimes I feel I can fight".More news: Kylie Jenner Ditches Lip Filler After Years Of Having Plumper Pout
Serena Williams has warned her Wimbledon rivals she is nowhere near her best despite powering into the quarter-finals without dropping a set.
Federer needed just 16 minutes to win the first set and the Swiss never looked back.
She steadied herself, though, and had a break point when Williams served at 4-2, but her chances were few and far between as her opponent continued to charge through a draw blown wide open by an exodus of seeds.
Williams is seeded 25th after returning from having a baby, but is looking like her usual dominant self on the grass courts.
"I feel like I'm getting to where I want to be".
In March, she lost in the first round of the Miami Open to Naomi Osaka, leading to a hasty exit in which she bypassed a postmatch media session. "But I feel like I'm back". No. 14 Daria Kasatkina, 2017 French Open champion Jelena Ostapenko vs. Dominika Cibulkova, and No. 13 Julia Goerges vs.
Kiki Bertens took care of the last of the top 10 with a 6-3, 7-6 (7/1) victory over Czech seventh seed Karolina Pliskova.
Ostapenko reached the last eight for the second straight year with a 7-6 (4), 6-0 victory over Aliaksandra Sasnovich, while Cibulkova beat Hsieh Su-Wei 6-4, 6-1.