The British number two had never previously won a main draw match at Flushing Meadows and looked to have a very tough draw on her hands against Russian Kasatkina, who is ranked ninth after winning a WTA Tour title on Saturday.
But Dart battled superbly in hot conditions on Court 12, recovering from a shaky second set and then coming from behind in the third to claim a 7-6 (8) 1-6 6-3 victory.
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Dart, 26, is enjoying the best season of her career having made the jump into the top 100 and picked up some eye-catching wins, including over Elina Svitolina on her way to the fourth round in Indian Wells.
She also had a strong grass-court season but the Londoner had never previously defeated a top-10 player.
There were 10 breaks of serve in the opening set, which Dart edged on a tie-break after saving two set points.
Kasatkina, who reached the French Open semi-finals earlier this year, hit back in the second set, pushing Dart well behind the baseline, and looked in control early in the third.
But Dart kept fighting and earned her reward, holding her arms aloft after clinching her first match point to set up a second-round clash with Hungary’s Dalma Galfi.