The third-seeded American fired an ace on match point to complete the comeback, which took 2 hours and 46 minutes on a hot and humid day on Arthur Ashe Stadium.
"I wasn't playing my best," the 25-year-old said in an on-court interview.
"It was super hot and that was no fun at all. I was sweating a lot," she added. "But I came out in the third set bouncing around."
The 21-year-old Kalinina blitzed Stephens at the start, racing out to a 4-1 lead in their first meeting before capturing the first set to make the partisan New York crowd nervous.
Stephens bounced back in the second, jumping out to a 3-0 lead before taking a medical timeout to receive treatment for a blister on the palm of her right hand.
The bandage on her serving hand seemed to hamper her and Kalinina broke Stephens in the first game after the timeout and levelled the set at 5-5.
But Stephens regrouped and crushed a running forehand winner to level the match and force a decider.
Stephens emerged from the 10-minute heat break following the second set the fresher of the two players, committing eight unforced errors to Kalinina's 17 as she cruised to the finish line.
Next up for Stephens is hard-hitting former world number one Victoria Azarenka of Belarus, who thrashed Australian Daria Gavrilova 6-1 6-2.
Despite the marathon match on Wednesday, Stephens is battle tested in New York.
She won four three-setters en route to last year's final, where she defeated fellow American Madison Keys in the straight sets to claim her first Grand Slam trophy.
(Reporting by Rory Carroll, editing by Ed Osmond)
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