Stearns was on the brink of the biggest win of her career when she led the reigning two-time Australian Open champion 5-4 40-0 in the third set with most of the 16,000 fans packed into Stadium Court behind her.
But the world number two came up with some gutsy returns to stay in the match and, even after appearing to hurt her ankle late in the final set, stayed aggressive in the tiebreak to book a third-round meeting with Emma Raducanu.
"Wow, she's an unbelievable player and she played crazy tennis," Sabalenka said of Stearns.
"At the net she said, 'I had you,' and she's right. I was probably just a little bit lucky to get this win. Super happy to get through this tough match."
Earlier, Coco Gauff overcame a 5-2 third-set deficit against unseeded Frenchwoman Clara Burel to emerge with a hard-fought 2-6, 6-3, 7-6(4) victory to reach the third round.
Third-seeded Gauff came out tight, spraying errors around the court and double faulting on two break points in a poor first set that Burel wrapped up with an unreturnable serve.
The U.S. Open champion stepped up her serving in the second set and moved into the court to take time away from her opponent, using her speed to race down a drop shot and level the contest at a set apiece.
Burel raced out to a 4-0 lead in the third set, but serving for the match up 5-3, it was 44th-ranked player's turn to tighten up, and Gauff pounced on her short balls and tentative serves.
Gauff kept her foot on the gas in the tiebreaker, sealing the win after a long rally ended with Burel misfiring wide, to the delight of the crowd on centre court.
"I'm happy with the mental fight," Gauff said. "It wasn't my best tennis, but it's not about how you show up on your good days, it's how you show up on your bad ones and I'm happy with how I showed up today."
Gauff will next face Lucia Bronzetti after the Italian beat Anhelina Kalinina 6-3 6-4.
PEGULA UPSET
Gauff's doubles partner and fifth seed Jessica Pegula was unable to stage a comeback of her own against Anna Blinkova, falling 6-2, 3-6, 6-3.
Pegula had beaten Blinkova in three sets at the San Diego Open last week, and it was the Russian who returned the favour thanks to some outstanding serving.
Former U.S. Open champion Raducanu advanced to the third round after Dayana Yastremska was forced to retire with an injury from their match while trailing 4-0 in the first set.
The pair exchanged a hug at the net and the British wildcard applauded the 30th-seeded Ukrainian as she left the court.
Wimbledon champion and seventh seed Marketa Vondrousova also withdrew from the tournament on Saturday, citing personal reasons, sending Ukrainian Marta Kostyuk into the fourth round.
Naomi Osaka continued to build positive momentum as the 2018 champion dispatched 14th seed Liudmila Samsonova 7-5, 6-3 behind her powerful ground game.
A beaming Osaka waved to the crowd after sealing the win. The Japanese player, who returned in January after a maternity break, will next face 24th seed Elise Mertens in the third round.
Ninth seed Maria Sakkari, 11th seed Daria Kasatkina, 16th seed Elina Svitolina and American Sloane Stephens were among the other players to advance to the third round with wins on Saturday.
