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Swiatek had not dropped a set at the season-ending tournament but was subdued by the powerful forehand of the Belarusian.
She outlasted Kasatkina in a 32-point rally for match point.
Maria Sakkari broke twice for a 5-1 cushion en route to capturing the first set, which meant Tunisian world number two Jabeur had no mathematical chance of advancing even if she had rallied back to victory.
Swiatek, who improved to 2-0 in this year's competition following her own 6-3, 6-2 win over Caroline Garcia, goes through as the winner of her group, while Gauff can no longer qualify for the semifinals.
The win continues a phenomenal season for Swiatek, who rose to world No. 1 in April and has won eight titles, including two Grand Slams at Roland Garros and Flushing Meadows this year.
Sakkari battled past world No. 3 Pegula 7-6(6) 7-6(4) to record her first win over a top-five opponent this year.
It was Pegula's 41st match victory in WTA main draws this year, with only world No. 1 Iga Swiatek (62) and the second-ranked Ons Jabeur (46) winning more.
She will join Iga Swiatek, Ons Jabeur, Jessica Pegula, Coco Gauff, Caroline Garcia, Aryna Sabalenka and Daria Kasatkina at the season finale.
'I struggled to win my first WTA title. It took me time. So I believe this will take me time,' said the Tunisian.
It was twice French Open champion Swiatek's first major title on a hard court and the 21-year-old is the first Polish woman to win the US Open.