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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.
Sakkari is the first player from her group through to the last four.
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.
Max Verstappen led from pole position at the Autodromo Hermanos Rodriguez, taking the chequered flag 15.186 seconds clear of Mercedes's Lewis Hamilton.
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.
The title, clinched with three races to spare, ended an eight-year streak by Mercedes who have yet to win a race this season.