Williams, after dropping a set at the All England Club for the first time in this year’s tournament, rallied for a 3-6, 6-3, 6-4 win on Center Court.
Williams, 36, who won the title in 2016, was absent from the All England Club in 2017 because of her pregnancy. After giving birth, she returned to play her first Grand Slam event at the French Open last month, reaching the fourth round.
The 52nd-ranked Giorgi, who had never taken a set off Williams in three previous meetings, overpowered the American to take the first set, hitting more winners and aces.
Williams, however, lifted her intensity in the second set, avoiding any break points as she secured one break of her own.
She now has a 39-35 record when losing the first set at a major.
Angelique Kerber, the highest remaining seeded player in the women’s draw at No. 11, finally converted on her seventh match point to advance to the Wimbledon semifinals.
The two-time Grand Slam champion from Germany defeated No. 14 Daria Kasatkina 6-3, 7-5 on Centre Court to reach the semifinals at the All England Club for a third time.
Kerber took advantage of Kasatkina’s erratic serving and errors to return to the last four. Kasatkina finished with seven double-faults, including one on break point in the second game and then two straight to hand Kerber a 5-3 lead in the first set — halting her momentum just after breaking to get back on serve with the help of one of the best rallies of the match.
Still, Kerber had some trouble closing things out.
She served for the victory at 5-4 in the second set, but got broken. When she served for it a second time, she needed to navigate a 16-point game that included five deuces and all of those match points, until forcing a forehand error on the last.
Kerber was the runner-up at Wimbledon in 2016, when she surged to No. 1 in the world rankings after winning the Australian Open and US Open. However, she took a big step back in 2017, going 1-12 against WTA top-20 players and failing to win a singles title.
She has bounced back in 2018, winning more than 75 percent of her matches and a title while racking up 10 wins vs. top-20 players.
Kasatkina, who earlier this year became the fifth player in WTA history with multiple wins over a WTA No. 1 before her 21st birthday, had 31 unforced errors, including seven double-faults. She was in the last eight at Wimbledon for the first time.
Kerber and Kasatkina are the only two women to reach the second week at all three majors this year.
In a match that featured eight breaks of serve — four in the first five games — Ostapenko was able to elevate her game in the crucial moments.
She was the aggressor throughout, hitting 33 winners to Cibulkova’s six, but also doubling her opponent’s unforced error count.
Goerges, who had never made it past the third round in her 10 previous appearances at the All England Club, advanced with a comeback victory over Bertens.
Goerges had gone out in the first round at Wimbledon for the past five years.
With Kerber also advancing, it is the first time two German women will play in the Wimbledon semifinals since 1931. The last time two German women reached any Grand Slam semifinals was at the 1993 French Open, featuring Steffi Graf and Anke Huber.
The Associated Press contributed to this report.