The 37-year-old ninth seed, who has already played two grand slam finals this season, blossomed under pressure in a nail-biting third set to remain on course for her first major title since she won Wimbledon in 2008.
Kvitova, back to her best after a career-threatening injury sidelined her for five months, was up a break in the decider but the 13th seed allowed Williams back into the contest and the American was the more composed player in the tiebreak.
"I have to say I felt every single one of you guys behind me. That feels good and I did not want to let you guys down," Williams told the 23,000 crowd on Arthur Ashe Stadium, before adding a word for her opponent, who was stabbed in her left hand by an intruder at her home last December.
"Everything she's gone through ... It's unbelievable. It's wonderful to see her back and play amazing. I'm so fortunate to have won that match."
Williams and Stephens could be joined in the last four by compatriots Coco Vandeweghe and Madison Keys, who are both playing on Wednesday.
Kvitova, whose previous five matches against Venus all went to three sets, led early on when a third Williams double fault in the third game put the Czech 2-1 up.
She still had not found her range properly, though, and Williams levelled for 3-3 with a break when Kvitova fired a forehand long.
Williams held serve in the next game and went on to set up three break points as Kvitova sprayed the court with unforced errors, converting the first with a neat backhand winner down the line.
The 2000 and 2001 U.S. Open champion continued to serve almost perfectly and pocketed the opening set when Kvitova, who hit 16 unforced errors, returned long.
Kvitova, whose backhand return was proving tough to handle, again went ahead in the second set with a break courtesy of a winner down the line and she held for 3-0 after saving break points.
The roof was closed as drops of rain started to fall and, after that brief interruption, Kvitova saved two break points before levelling the tie with a big first serve that Williams returned long.
Kvitova also drew first blood in the decider, using her crafty backhand to devastating effect to break for 2-1, but a double fault on break point in the sixth game allowed Williams to level.
The match always looked like being decided in the tiebreak and Williams took an early mini-break before dictating play with deep forehands to take a 6-1 lead.
She double faulted on her first match point but Kvitova returned wide on the second.
(Reporting by Julien Pretot; Editing by Nick Mulvenney)
