Giorgi was a 6-3 6-4 victor against Ekaterina Makarova and Goerges beat Donna Vekic 6-3 6-2.
Ominously for the other seven women left in the draw, the 23-time Grand Slam champion, who missed Wimbledon a year ago, but won the title on her previous two visits, insists there is plenty to improve on.
Williams took control of the third and final set in the the third game, claiming the third break of the match, and with the remaining games all going to serve, served out for a 3-6, 6-3 6-4 win.
Serena Williams loses the first set, but recovers to defeat Camila Giorgi and continue her quest for an eighth Wimbledon title.
"Even when I was down the first set, I thought she's playing great". This time around focus was on whether Williams would be seeded (she was) and where she would land up in the draw. But the 36-year-old Williams, back at Wimbledon after missing last year when she was pregnant with her first child, dug in, upped her game and reeled Giorgi in. It's like today, I hope I get off to a good start, go from there.
"Sometimes I feel, 'Man, I'm in trouble, '" she said after the match.More news: Canada offers to lead new NATO operation in Iraq
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She will now face Julia Goerges for the second time in as many Slams, but it was during that match that she picked up the injury that cut short her Slam come-back, But today we saw a return to the serving speeds of 122mph.
After going a couple of weeks without hitting a serve, Williams has regained her ability with that stroke nicely at Wimbledon.
Williams will face Italian world number 52 Camila Giorgi in the last eight, with the victor going on to face either Bertens or Germany's Julia Goerges in the semi-finals. I played some great matches, and these games will help me for the rest of the year. Pliskova had been the only female player in the top 10 to reach the fourth round, the fewest in a Grand Slam event in the Open era following a slew of upsets in the first week. In Giorgi, Williams was facing an opponent who was young (26), strong and playing, perhaps, the best tennis of her career.
Then over the weekend, Serena revealed she had missed Olympia walking for the first time while she was away at a practice session.
"We both played at a really high level, starting from the first point".
The All England Club seeded her there as a nod to all of her past success at the grass-court major, including titles the last two times she entered, in 2015 and 2016.
Dominika Cibulkova, whose straight-sets win over Hsieh Su-wei was overshadowed by a lengthy argument with umpire Zhang Juan over a disputed point, faces last year's French Open victor and 12th seed Jelena Ostapenko.