“Mentally, I am happy that I stayed focused. I got some confidence in the second set and everything was going my way,” Rybakina said.
“Hopefully I will play like that through the whole tournament.”
Anisimova battled back from a set and a break down to beat compatriot Keys 4-6 6-3 6-2 and set up a winner-takes-all final group match against Poland’s Swiatek on Wednesday.
Wimbledon and US Open runner-up Anisimova, appearing at her first Finals, let out a roar of frustration after going 2-0 down in the second set but recovered superbly to win 12 of the next 15 games.
Keys did not shake hands with Anisimova at the end of the match, explaining to her opponent that she was ill.
The leading eight women’s players in singles and doubles are competing in Saudi Arabia, where £12m ($15.5m) in prize money will be won.