Winless in three previous meetings with Williams, including last month in Paris, Goerges could face compatriot Angelique Kerber in the first-ever all-German women's grand slam final if she somehow fashions a way to turn the tables.
However, Williams lifted her intensity to level, before closing out the match to move a step closer to reclaiming the title she won in 2016.
"Even when I was down the first set, I thought, "Well, she's playing great".
She'll face Germany's Julia Gorges in the semifinals on Thursday at the All England Club.
Men's quarterfinals: Kevin Anderson beat Roger Federer 2-6, 6-7 (5), 7-5, 6-4, 13-11; John Isner beat Milos Raonic 6-7 (5), 7-6 (7), 6-4, 6-3; Novak Djokovic beat Kei Nishikori 6-3, 3-6, 6-2, 6-2; Rafael Nadal beat Juan Martin del Potro 7-5, 6-7 (7), 4-6, 6-4, 6-4.
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Williams needs to beat two seeded opponents to win her 24th major title, tying Margaret Court's record for Grand Slam wins.
She served for the victory at 5-4 in the second set, but got broken.
Williams is chasing an eighth Wimbledon crown and the American star on Monday showed why she is the title favorite with a 6-2, 6-2 demolition of Evgeniya Rodina in the fourth round.
"To be here, to be in the semi-finals". Giorgi reduced the tally but the set proceeded with serve to end 6-3 for Serena. "For whatever reason, I was so calm", Williams said.
"I'm going to go out there and just do what I can, do my best, see what happens".
Angelique Kerber and Jelena Ostapenko to meet in Wimbledon last four
Goerges also battled back from a set down to beat Dutchwoman Kiki Bertens 3-6 7-5 6-1 in the last of the women's quarter-finals. By the time she flubbed a backhand while falling behind 5-1 in the second, she dropped her racket and screamed.
Pliskova was soon again presented with opportunities to even the set, but this time Bertens fended off the Czech's attack, saving all four for a 5-3 lead. Four of the first five games went against serve before 12th seed Ostapenko's game finally caught fire in the 11th game when she broke Cibulkova to love.
With no top-10 seeds making the quarterfinals for the first time in Wimbledon history, 11th-seeded Kerber is the highest-ranked player left in the hunt for the Venus Rosewater Dish. "For me, there's so much farther I want to go to get back where I was - and hopefully go beyond that".
"It's pretty unreal for me", said Goerges, who reached her first major semifinal at a tournament where she exited in the first round each of the past five years.
"It was great matches", Giorgi said in her postmatch news conference. For the first time at this year's Wimbledon, she had lost a set. Both two-time Grand Slam champion Kerber and 21-year-old Kasatkina reached the quarterfinals at last month's French Open.
"Wow, it's pretty wonderful", Goerges said. Meanwhile, Ostapenko also achieved a win in straight sets against Dominika Cibulkova, 7-5, 6-4.
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