Australian Open champion and top seed Rybakina was forced to call it a day in her last-16 match due to gastrointestinal issues at the start of the third set against Croatian lucky loser Antonia Ruzic.
The Kazakh joins other high-profile players, including the world's top two Aryna Sabalenka and Iga Swiatek, as well as Victoria Mboko and Barbora Krejcikova, in retiring or withdrawing from the WTA 1000 event.
Ruzic, who in the first round eliminated 2021 US Open winner Emma Raducanu, has reached the first WTA 1000 quarter-final of her career.
"It was actually a crazy story that I got in as a lucky loser because I lost the first round of qualification," Ruzic said.
"A lot of girls had amazing runs from the lucky losers because you get a second chance, so you have nothing to lose anymore. You can just show your best tennis. I'm happy with how I played today."
The 23-year-old world number 67 will face seventh seed Elina Svitolina for a spot in the last four after the Ukrainian beat Belinda Bencic 4-6, 6-1, 6-3.
Anisimova, the second seed, played her first match of the tournament after getting a bye through the first round and a walkover in the second as she beat Janice Tjen 6-1, 6-3 to book a last-eight spot.
There she will meet defending champion Andreeva, who defeated Jaqueline Cristian 7-5, 6-3.
It was also the 18-year-old fifth seed's first match of her title defence after similarly receiving a bye and a walkover earlier in the week.
"Honestly, I feel my game was really far from perfect today," Andreeva said.
"I was tense, I was nervous, I cared a lot about every single point that I was playing and sometimes I was really scared to miss.
"Tomorrow (Thursday) is probably going to be better, because I'm going to get into the rhythm... I'm looking forward to getting that revenge!"
Andreeva's only previous meeting with Anisimova ended in defeat in the Miami Open third round last year.
Third seed Coco Gauff saved three match points as she fought back to win 2-6, 7-6(9), 6-3 against Elise Mertens and will meet either Alexandra Eala or Sorana Cirstea in the next round.
American Jessica Pegula eliminated her 18-year-old compatriot Iva Jovic 6-4, 6-2 and will next face Clara Tauson following the Dane's victory by the same scoreline against Magda Linette.
