NEW YORK – For eight intense, entertaining games, Sloane Stephens stayed right with Serena Williams.

For a 40-minute stretch in the fourth round of the U.S. Open, the top two American women put on quite a show. Both hit serves topping 115 mph. Both whipped groundstrokes to the corners. Both covered a lot of ground, extending points with leg-churning defense. Both showed the occasional sign of nerves, reflecting what a big deal this was, in part because the 15th-seeded Stephens already was one of only three players to beat the No. 1 Williams this season.

Until, that is, the score was 4-all in the first set Sunday. That’s when Williams took over.

The 20-year-old Stephens’ time at the top of tennis may come. For now, the 31-year-old Williams is still as good as it gets. Taking eight of the last nine games, defending champion Williams returned to the quarterfinals at Flushing Meadows by beating Stephens 6-4, 6-1.

“When you give her that opportunity to take that step forward, she definitely makes her move,” Stephens said. “Unfortunately, today she made her move. I just couldn’t get back in.”

Still, all in all, it was remarkably compelling and, within individual points, rather evenly played for what turned out to be such a runaway.

“I definitely think it was a high-quality match,” said Williams, 64-4 with eight titles this year. “We both came out today to play.”

She advanced to play No. 18 Carla Suarez Navarro of Spain, who defeated No. 8 Angelique Kerber of Germany 4-6, 6-3, 7-6 (3).

Meanwhile, No. 3 Agnieszka Radwanska suffered a 6-4, 6-4 loss to 24th-seeded Ekaterina Makarova of Russia. Next for Makarova is a match against fifth-seeded Li Na of China, the 2011 French Open champion, who eliminated 2008 U.S. Open runner-up and No. 9-ranked Jelena Jankovic 6-3, 6-0 in less than an hour Sunday night.

In men’s third-round action, the last of 15 Americans in the field lost, making this the first U.S. Open in history without at least one representative from the host country in the fourth round. Wild-card recipient Tim Smyczek, ranked 109th, was beaten 6-4, 4-6, 0-6, 6-3, 7-5 by Marcel Granollers of Spain.

It also means no U.S. man reached the fourth round at any of the four Grand Slam tournaments in 2013.

Granollers, who is ranked 43rd, next takes on No. 1 Novak Djokovic, a 6-0, 6-2, 6-2 winner over 95th-ranked Joao Sousa of Portugal.

The man Djokovic lost to in last year’s U.S. Open final and this year’s Wimbledon final, Andy Murray, struggled with his breathing on a muggy afternoon but otherwise faced little trouble in a 7-6 (2), 6-2, 6-2 victory over 47th-ranked Florian Mayer of Germany. Murray has won 29 of his last 31 Grand Slam matches, a run of success that includes his first two major titles — at the U.S. Open last September and Wimbledon this July — along with two runner-up finishes.

“The expectations are higher, but there’s not as much pressure to win,” the third-seeded Murray said.

“I feel much more comfortable coming into these events than this time last year.”