It had the makings of a delicious, competitive clash, world No. 1 Serena Williams against fast-rising challenger Sloane Stephens.

And for nearly an hour, the top two Americans in the women’s tennis rankings traded power for power Sunday with a spot in the U.S. Open quarterfinals at stake.

Just a few weeks shy of her 32nd birthday, Williams stared down the likely future of the women’s game in the United States and, with the force of her groundstrokes, the skilled placement of her serves and a fierce eruption of “C’mon!” upon getting a critical service break, declared she has no interest in being supplanted any time soon.

With her 6-4, 6-1 victory, the defending champion clinched a berth among the tournament’s final eight and exacted revenge against the 20-year-old Stephens, who ousted her from the Australian Open in January.

“Going into the match, I definitely wanted to be focused the whole time,” said Williams, who finished with 22 winners and 13 unforced errors. “Whether I was going to win or lose, I just wanted to play my game and do well.”

It was the third installment of their budding rivalry. Williams prevailed in straight sets in a tournament in Brisbane, Australia, in early January. Three weeks later, Stephens pulled off the three-set shocker in the season’s first major.

At the time Williams was hampered by a sprained ankle and back spasms.

Sunday’s match offered a truer picture of the competitive gap between them, with Williams seizing the momentum early in the second set and never relenting, managing to hike up her intensity and accuracy down the stretch while her young challenger seemed to lose her bearings and belief.

But if Stephens has proved anything this season, toppling Williams in January and third-ranked Maria Sharapova in August, it’s that she’s a quick study and is closing fast on the top ranks of the sport. She also competes with impressive composure, telegraphing little if any frustration when shots go awry. And she thrives on the sport’s biggest stages, having reached the fourth round or better at all four majors this season.

“I embrace it,” Stephens said when asked about being characterized so often as the “heir apparent” to Williams, who has won 16 majors, including four U.S. Open championships.

“Everyone has their time to shine,” added Stephens, who, like so many tennis-obsessed youngsters, once decorated her bedroom with a poster of Williams. “She’s obviously number one in the world for a reason. She has earned every opportunity she has gotten. I think she makes the most out of every time she’s on the court.”

Their relationship has been the topic of tremendous speculation.

After her defeat in Brisbane, Stephens was quoted as criticizing Williams for exulting too much on court.

Then in a May interview with ESPN the Magazine, Stephens said Williams wasn’t as friendly as most people believed, portraying her as cold and distant following Stephens’s victory at the Australian Open.

Stephens later said she hadn’t understood the comments would be publicly aired and insisted she had nothing but admiration for Williams, who singled out Stephens for praise heading into Wimbledon.

On the eve of Sunday’s match, Stephens referred to Williams as a “co-worker,” saying: “We’re Fed Cup teammates. But other than that, everything else is private.”

Williams, for her part, insisted she didn’t feel like a favorite heading into Sunday’s fourth-round encounter, calling Stephens “a great competitor — one of the best players to ever play the game.”

They were evenly matched for much of the first set at Arthur Ashe Stadium.

Williams blasted an ace out wide to start the proceedings and held serve at love.

Stephens replied with a tougher fought but impressive hold, easily handling Williams’s vaunted power and winning a 23-stroke rally.

While Williams held serve with ease, Stephens made her own statement of strength in fighting so hard to hold her serve in reply. She made Williams work on each rally, yanking her from side to side with sharply angled forehand blasts.

Serving at 4-5, Stephens fended off two set points before getting broken to lose the set.

Facing break point in the opening game of the second set, Williams uncorked two huge serves to dig out of trouble. Then she broke Stephens in the fourth game by playing terrific defense to take a 3-1 lead, then erupted with a huge “C’mon!” and fist pump.

Williams cruised from there as Stephens started over-hitting in an effort to stay in the match.

After Stephens’s final forehand plowed into the net to end the match, they met at the net and clasped hands. Moments later, Williams praised Stephens once more during an on-court interview broadcast to the crowd.

“Sloane is such a great player!” Williams said. “How excited are we for the future of American tennis, right?”

Indeed, Stephens did so many things right. She had no trouble with the pace of Williams’s serve. She played the aggressor at key moments. She ripped her forehand with authority. And she seemed the fitter woman, barely breathing hard after furious rallies.

But with her occasional wild shots, whether struck off balance or overly aggressive, she also demonstrated that the future of American tennis needs a bit more incubation time.

Notes: Earlier Sunday, defending U.S. Open men’s champion Andy Murray needed less than two hours to dispatch Germany’s Florian Mayer, 7-6 (7-2), 6-2, 6-2.

Top seed Novak Djokovic needed even less time against Joao Sousa of Portugal, advancing, 6-0, 6-2, 6-2, in 1 hour 41 minutes.

The women’s draw lost No. 3 seed Agnieszka Radwanksa, who was ousted by Ekaterina Makarova, 6-4, 6-4. And No. 5 Li Na dismantled Jelena Jankovic, 6-3, 6-0.

The top-ranked doubles team of Bob and Mike Bryan survived a scare but advanced to the quarterfinals upon defeating the unseeded tandem of Canadians Daniel Nestor and Vasek Pospisil, 6-7 (7-1), 7-5, 6-2.

The 35-year-old twins, who have won 15 major titles, are trying to become the first men’s doubles team in 62 years to win all four majors in the same year.