Potapova defended 13 of the 16 break points she faced to reach the fourth round in New York for the first time, where she will play either Russian Mirra Andreeva or Czech Nikola Bartunkova.
"I think I'm the happiest person out here - maybe my coach is a little bit happier," said Potapova.
"To play here gives me so many goosebumps when I come out on this court in front of this crowd."
Anisimova, a runner-up at Wimbledon last year as well, has been unable to reach a tournament final in 2026 and had a long chat with her coaches after the opening set, where Potapova broke her to love in the third game and converted on another break point chance in the seventh.
The pair traded breaks three times in a competitive second set, when the crowd at Louis Armstrong urged on the American at every turn.
But Anisimova, who had 46 unforced errors across the match, gave her opponent a leg-up when she sent a shot just past the baseline on break point in the penultimate game.
Potapova absorbed Anisimova's power from the baseline in a fine final rally, clasping her hands to her mouth after the American sent the ball out again on match point.
