World number five Jabeur, who had minor surgery in February and skipped events in Doha and Dubai, was trailing her Russian opponent 3-5 in the first set when rain stopped play and she returned rejuvenated to complete a 7-5 7-5 victory.
"I have to thank the rain," Tunisian Jabeur, the runner-up to Swiss Belinda Bencic (26) in 2022, told reporters on Saturday.
"It allowed me to talk to my coach a bit and he gave me some advice and I reflected on what happened in the first set.
"My game was there, I just needed to find something small. I was more patient definitely after, I was building my point more and that really helped me."
Kasatkina said she was frustrated by the delay which meant the match ended more than five hours after it began.
"I just wish all tennis was played indoors like most of the other sports," Kasatkina said.
Up next for Jabeur, last year's Wimbledon and U.S. Open runner-up, is either top seed Jessica Pegula (29) or holder Bencic.
The second semi-final was suspended late on Saturday with Bencic up 7-5 6-6 but Pegula ahead 4-2 in the tiebreak.
The match will be completed on Sunday with the final scheduled to take place later in the day.
