Four-time U.S. Open winner Williams grabbed the only service break of the first set in the sixth game and immediately seized the advantage by breaking the 78th-ranked Shvedova in the opening game of the second set and charged on to victory.
The routine win moved the world number one and top seed into the round of 16 against 20-year-old Stephens, who defeated Williams in the quarter-finals of this year's Australian Open.
The 15th-seeded Stephens defeated compatriot Jamie Hampton 6-1 6-3 in her third-round match.
Williams, 31, was never really threatened by Shvedova, who had advanced to the third round with the loss of just six games.
The 25-year-old with the heavy, flat groundstrokes produced only three winners as she fell short of becoming the first woman to reach the fourth round of the U.S. Open representing Kazakhstan.
Williams dismissed the notion that she would be playing for revenge against Stephens with a berth in the quarter-finals at stake.
"It's going to be tough, Sloane is playing so well," Williams said on court after her late finish in the final night match on Arthur Ashe Stadium.
"Regardless, there's going to be one American in the quarter-finals, and I'm really proud of Sloane and it's going to be a really good match."
Stephens, after her victory, said she was eager for a clash against Williams.
"I think it will be an epic," said Stephens. "I'm really looking forward to it."
The young American lost to Williams in the Brisbane quarter-finals two weeks before the Australia Open.
"She's a great competitor, one of the best players to ever play the game," she said about what she admires about the 16-times grand slam singles winner.
"She's a fierce competitor, obviously she's number one in the world for a reason. She's very aggressive. She stays on top of you, doesn't give you any room to breathe. She's intense. She knows what she wants to do out there.
"That's why she's number one in the world."
(Reporting by Larry Fine, Editing by Nick Mulvenney/Patrick Johnston)
