Anisimova saved two match points in the third set at Margaret Court Arena ahead of the tiebreaker and ended the match with an ace.
The Czech fourth seed was one of the standout players of 2021, winning three titles as she soared up the rankings to her current world number four.
Indian Wells 2021: Victoria Azarenka beats Jelena Ostapenko, books spot in final against Paula Badosa
Paula Badosa defeated Ons Jabeur in straight sets 6-3, 6-3 to reach the final.
Dimitrov will face Polish eighth seed Hubert Hurkacz, who rolled over Russian Aslan Karatsev 6-1, 6-3 earlier on Wednesday, in the quarter-finals.
Zverev advanced to the fourth round, coming from behind in both sets to beat two time Olympic gold medal winner Murray 6-4, 7-6 (7/4)
Jannik Sinner, ranked 15th, came good in Cincinnati as he defeated Argentine Federico Delbonis 6-2, 7-5.
Sakkari is the first Greek woman to get this far here. Her ailing opponent needed a medical timeout before quitting the match after 91 minutes on court.
Wimbledon 2021: Roger Federer glides into last-16, Ashleigh Barty polishes her act, Nick Kyrgios retires, Daniil Medvedev stages comeback win
Eight-time champion Federer triumphed 6-4, 6-4, 5-7, 6-4 and will be playing in the fourth round of a Grand Slam for the 69th time in his career.
Wimbledon 2021: Ajla Tomljanovic accuses Jelena Ostapenko of 'lying' about injury in heated third round clash
Tomljanovic was furious that 2017 French Open champion Ostapenko called the trainer at 0-4 in the final set, claiming she was suffering from an abdominal injury. Tomljanovic eventually clinched a 4-6, 6-4, 6-2 win.
Ranked 43 in the world, former French Open champion Ostapenko was handed a wildcard to the Wimbledon warm-up event and could be a contender at the All England Club where she was a semi-finalist in 2018.
Jelena Ostapenko to face Anett Kontaveit in Eastbourne final; Lorenzo Sonego vs Alex De Minaur for men
Former French Open champion Ostapenko, now 43rd in the world rankings, struck 38 winners against Kazakhstan's Rybakina, who appeared to be hindered late in the first set by a back problem.
Eastbourne defending champion Karolina Pliskova beaten by Camila Giorgi, Jelena Ostapenko, Elina Svitolina win
The fifth-seeded Czech was beaten 2-6, 6-2, 6-2 by Italian qualifier Giorgi in the first round of the Wimbledon warm-up event on England's south-coast.
On Court Philippe Chatrier, Federer kicked off only his second French Open campaign in six years with an impressive dismissal of Denis Istomin.
Swiatek, who turns 20 next week, was the lowest-ranked woman, then 54th in the world, to win the French Open, shifted from its traditional May-June slot last year due to the Covid-19 pandemic.
Italian Open 2021: Ashleigh Barty retires injured in quarter-final, Karolina Pliskova beats Jelena Ostapenko
Top seed Barty, the 2019 French Open champion, was leading 6-4, 2-1 when she pulled up to receive treatment during her match against American Coco Gauff.
Stuttgart Open: Karolina Pliskova beats Jelena Ostapenko to set up quarter-final against Ashleigh Barty
Pliskova, the sixth seed, recovered from losing the first set in a tie break by firing down 21 aces, 10 of which came in the last set alone, for a 6-7 (7/9), 6-4, 6-3 second-round win.
Latvia beat India 3-1 in the Billie Jean King Cup playoff with Ankita Raina once again highlighting her fighting spirit despite two defeats.
Latvia took a 2-0 lead over India in the Billie Jean King Cup playoff tie with wins for Jelena Ostapenko, Anastasija Sevastova over Ankita Raina and Karman Kaur Thandi respectively.
India take on higher-ranked Latvia in their first-ever Billie Jean King Cup playoff tie. The matches will be played on indoor hard courts in an empty stadium.
Billi Jean King Cup: We are hungry, upsets do happen in team events in tennis, says India captain Vishal Uppal ahead of tie against Latvia
Asked about the approach the team will adopt, considering their superior rivals, Uppal oozed confidence.