Ninth seed Williams and Stephens qualified with 6-3 3-6 6-1 wins against gritty Spaniard Carla Suarez Navarro and German Julia Goerges, respectively.
They could be joined in the last eight by Jennifer Brady, CoCo Vandeweghe and Madison Keys, who are all in fourth round action on Monday.
Both Williams and Stephens are enjoying comeback tales.
Williams made the Australian Open and Wimbledon finals this year, her first grand slam championship matches since 2009, while Stephens is back to form after almost a year out following foot surgery.
"I think that now I'm running around and sweating, my ponytail is flying, these are all such great things," said Stephens.
The 24-year-old entered the temporary structure of Louis Armstrong Stadium with a 4-1 career record against 30th seed Goerges.
She wasted three set points at 5-3 in the opening set but in the fourth, her opponent hit wide and Stephens seemed in control. But she lost her focus and the German levelled a contest of baseliners.
Goerges couldn't keep up with the pace in the decider, though, and Stephens wrapped it up comfortably.
The same scenario applied on Arthur Ashe Stadium, where Williams, with the backing of the 23,000 crowd, raced through the first set before Suarez Navarro's sliced backhand started to trouble her.
The Spaniard's effort to level the match cost her, however, and Williams played with more accuracy in the decider as Suarez Navarro grew frustrated before capitulating.
A U.S. Open champion in 2000 and 2001, Williams has been around long enough to deal with her opponents' aggression.
"The depth in the game is so much deeper now than it was at the time," Williams said.
"It's never easy to win a slam. I mean, now I'll play players who I've never seen or know their name, and you can't let up one point.
"Players come out with a lot more confidence now. Every time I step out on the court, that person is coming for me.
"It doesn't matter if I played for a year, not even if I'm ranked 100 or number whatever, they're coming for me.
"When I get to these majors, I know that's what it is."
(Reporting by Julien Pretot; Editing by Ian Ransom)
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