Rodina, who like Williams combines a tennis career with motherhood, looked overawed at first as she went 3-0 down.
Bidding for an eighth championship at the All England Club, Williams will be back on court Tuesday, but she is seeded only 25th, and that actually represents a big bump up from her ranking of No. 181. With the top 10 women having all been eliminated, No 11 Angela Kerber is the highest-seeded player left in the tournament.
Against Bertens, she won 3-6, 7-5, 6-1 in one hour and 57 minutes.
Goerges will face seven-time champion Serena Williams in the semi-finals on Thursday.
The last time this occurred at a major was Wimbledon 2016, which featured Serena Williams, Angelique Kerber and Venus Williams. As she keeps reminding folks, this is just her fourth tournament since missing more than a year of action because she was pregnant and gave birth to a daughter last September.
"I faced a thousand and three seeds in my life, so I'm okay with that", Williams responded when asked if she would have preferred stiffer tests.
Ostapenko was the aggressor throughout, hitting 33 winners to Cibulkova's six, but also hitting 28 unforced errors to 13 for her opponent.
It took Mannarino so much toil and trouble to win one game, saving four break points to hold to begin the second set.
Goerges had suffered five successive opening round defeats at the All England Club before this year.
Giorgi, Kasatkina, Bertens and Goerges will each be making her debut in the Wimbledon quarterfinals.
Angelique Kerber reached her third Wimbledon semifinal after converting her seventh match point to beat Daria Kasatkina 6-3, 7-5 on Tuesday, setting up a meeting with former French Open champion Jelena Ostapenko.
Ranked at world No. 181, Serena was the lowest-ranked player to contest a last-eight match in these parts in the Open Era, although she is the 25th seed. "But yeah, every single match I play, whether I'm coming back from a baby, or surgery, it doesn't matter, because these young ladies bring a game that I've never seen before ..." "Everybody has the same chances to win that match, and I'm looking forward to it". Del Potro was leading Simon two sets to one when their match was suspended because of darkness Monday night.