The stage was set for the highly anticipated clash between the pair after Swiatek routed Anisimova 6-0, 6-0 to lift her maiden Wimbledon crown in July, before the American gained revenge in the US Open quarter-finals in September.
Following a fierce start to their latest meeting and 12 straight holds of serve, Poland's Swiatek raised her level to seize control of the first set in the tiebreak, clinching it when Anisimova struck a forehand long.
The war of attrition continued in the second set, before Anisimova ramped up the intensity on her forehand and earned the first break of the match in the 10th game to force a decider at the King Saud University Sports Arena.
With confidence flowing, Anisimova carved out another break to surge 3-1 ahead in the third set and shift the pressure onto world number two Swiatek, and the resilient 24-year-old stayed firm from there and completed her comeback victory.
Dominant run
Rybakina earlier continued her dominant run with a 6-4, 6-4 win over second alternate Ekaterina Alexandrova, after Madison Keys withdrew from their clash with illness, and Mirra Andreeva was not fit to play.
"It was tough, especially in the second set when I was up and I lost a bit of concentration," said Rybakina.
"Ekaterina is always a tough opponent, she also has a big serve. In the beginning it was a bit difficult to return, so I'm pretty happy that I managed to win in two sets in the end."
Rybakina arrived for the clash after victories over Amanda Anisimova and Iga Swiatek, but the former Wimbledon champion came under pressure against a fresh Alexandrova, before breaking in the ninth game and then claiming the opening set with a hold.
The Kazakh, who staved off three breakpoints in the first set, cranked up her big serve and heavy hitting early in the second to go 3-1 up and applied the squeeze to close in on another win ahead of the semi-finals.
Another break courtesy of a powerful return of serve left Rybakina one game away, and the 26-year-old eventually wrapped up the victory despite some struggles on her own delivery late in the clash.
World number one Aryna Sabalenka takes on Coco Gauff on Thursday with both players in contention for a semi-final place from the Steffi Graf group while Jessica Pegula can boost her own chances when she meets the eliminated Jasmine Paolini.