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.
"Every time I play Giorgi she always plays that level so I knew going in it would not be an easy match", Williams, who will face German 13th seed Julia Goerges in the semi-finals, said shortly after walking off the court.
"I faced a thousand and three seeds in my life, so I'm okay", she said.
Serena Williams figures this statistic needs an asterisk: For the first time since Wimbledon started seeding players in the 1920s, none of the top 10 women reached the quarterfinals. "I felt like I did some good business here in the past already", eight-time champion Federer said on whether he still has unfinished business at Wimbledon.
Williams bounced back in the second, breaking in the fourth, from which point she never looked back, further holds of serves from both players seeing the American take it 6-3.
Serena Williams and her daughter Alexis Olympia watch the action during the first round of the 2018 Fed Cup at US Cellular Center on February 10, 2018 in Asheville, North Carolina.
Julia Goerges of Germany celebrates breaking the serve of Kiki Bertens of the Netherlands during their women's quarterfinal match at the Wimbledon Tennis Championships in London, Tuesday July 10, 2018. She missed Wimbledon a year ago because she was pregnant, going about 16 months between Grand Slam tournaments, so her ranking is just outside the top 180.
"I said, 'All right, ' " said Williams, who hadn't dropped a set the entire tournament before Tuesday. Camila served brilliantly to reduce the tally but the Serena's unrelenting attack sustained the gap to 5-3. 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.
Having shaken off the rust following her lengthy lay-off after the birth of her daughter Olympia in September, Williams is on the verge of a 10th Wimbledon final appearance.
"She is clearly a really good grass-court player. I still have a long way to go to be where I was".
"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.
Cibulkova did her best to try and hang on but the Slovakian was left chasing shadows as Ostapenko blasted a backhand victor on her first match point to reach the last four without dropping a set in this year's tournament.
Bertens was bidding to become the first Dutch woman to reach a Wimbledon semi-final in over 40 years and took the opener in emphatic style. I was not thinking too much about the match points. It appeared nearly to be a reminder to Giorgi, who said on Monday that she doesn't watch women's tennis. She hasn't had to try very hard to win her points against an error-prone Ostapenko.