Sabalenka, ranked 78th, took the game to Stephens in a gripping contest of contrasting styles, and it was the Belarusian teenager's aggressive all-or-nothing tennis that won the day against the more defensive American.
World number 10 Vandeweghe put in a composed performance against Aliaksandra Sasnovich to send the visitors ahead with a 6-4 6-4 victory.
Vandeweghe faces Sabalenka and Stephens meets Sasnovich in the reverse singles on Sunday as the U.S. look to get back on track in the hunt for an elusive 18th title. The U.S. last won the Fed Cup 17 years ago while Belarus are playing their first final.
Sabalenka stormed out of the blocks, breaking Stephens in the opening game and racing into a 3-0 lead as the world number 13 struggled to contend with the 19-year-old's powerful groundstrokes.
Stephens bounced back in the second set using the strategy that served her well at Flushing Meadows - getting balls back into court and counting on Sabalenka making unforced errors.
The plan worked and the match went to decider as Sabalenka kept going for her shots and missing.
Both players exchanged breaks in the third before Sabalenka held with an ace to go 4-3 up and the momentum shifted back to her.
She sealed victory on Stephens' serve, dropping to her knees after the American sent a forehand into the net.
"I had to win, I had to make the score 1-1, that's all I had in my mind during the match," Sabalenka said. "It's the first time that I've cried after a match.
"It would have been really hard tomorrow if the score was 0-2 and now it's 1-1 so there's a chance for us to win."
Vandeweghe started well against 87th-ranked Sasnovich, breaking in the first game and holding serve with ease.
She broke again in the first game of the second set, and although Sasnovich levelled things up immediately, Vandeweghe struck back before holding to love to take control.
She briefly lost her composure on serve with the score at 4-3, surrendering a surprise break after leading 40-0, but broke back immediately to restore her advantage before serving out for victory.
(Writing by Simon Jennings in Bengaluru, editing by Ed Osmond)
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