* Serena outguns Stephens in all-American battle
* Murray overcomes steamy conditions for straight-sets win
* Djokovic yet to drop a set
* Former champion Hewitt continues his run (Updates at end of day)
By Julian Linden
NEW YORK, Sept 1 (Reuters) - Serena Williams easily won her eagerly-awaited clash with Sloane Stephens at the U.S. Open on Sunday as normal service resumed at the last grand slam of year.
Williams was one of four U.S. Open champions that won on Sunday, sending a clear message to the next generation of rising stars that they will have to wait their turn.
The 31-year-old avenged her loss to Stephens at the Australian Open with an emphatic 6-4 6-1 victory to keep her title defence on course.
"It's definitely difficult, especially playing people that you like, that you always want to see do well," Williams said.
"But you have to go out there and kind of put that to the side and realize, I want to do well myself and take every point as it comes."
Defending men's champion Andy Murray and world number one Novak Djokovic also won in straight sets, while 2001 winner Lleyton Hewitt turned back the clock to reach the fourth round for the first time since 2006.
Murray, playing in the middle of a hot and steamy day, needed less than two hours to see off Germany's Florian Mayer 7-6 (2) 6-2 6-2 and advance to the fourth round.
Djokovic was even more ruthless in his center court night match against Portugal's Joao Sousa, cruising to a 6-0 6-2 6-2 win, while 32-year-old Hewitt wore down Russia's Evgeny Donskoy 6-3 7-6 (5) 3-6 6-1.
"Obviously when you've been to the top you want to keep playing," said Hewitt, who stunned sixth seed Juan Martin del Potro in the previous round.
"The reason you're playing is for the majors, and for me, Davis Cup as well. That's the reason I'm still playing."
The biggest surprise on a day when most results went as expected was the defeat of women's third seed Agnieszka Radwanska.
The Pole, who had made it to the quarter-finals or better at the three other grand slams this year, crumbled to a 6-4 6-4 defeat by Russia's Ekaterina Makarova after she had won the first four games of the match.
Makarova wil play China's Li Na after Asia's top player accounted for Jelena Jankovic 6-3 6-0.
Stephens beat Williams in January in the quarter-finals at Melbourne Park to enhance her standing as a future leader of U.S. women's tennis, but was completely outgunned by a woman bidding to become the oldest to win the U.S. Open since tennis turned professional in 1968.
"Obviously she's number one in the world for a reason," said Stephens, 11 years younger than Williams. "I thought she played really well.
"All in all I thought I competed well and played well. That's all you can do really."
Williams will play Carla Suarez Navarro in the quarter-finals, the Spaniard having upset Germany's eighth-seed Angelique Kerber 4-6 6-3 7-6 (3).
Murray struggled to impose himself on Mayer in the opening set but once he adapted to the humidity and gusting winds it was all smooth sailing.
"It was tough conditions. It was very, very humid," Murray said. "I was struggling breathing for most of the match."
Murray's opponent in the fourth round will be Uzbekistan's Denis Istomin, who beat Italy's Andreas Seppi 6-3 6-4 2-6 3-6 6-1, with Tomas Berdych looming in the quarter-finals.
Berdych, seeded fifth, made the semi-finals at last year's U.S. Open and has been in great form at Flushing Meadows this week, reaching the last 16 without dropping a set.
The Czech dispatched Julien Benneteau of France 6-0 6-3 6-2 to set up a fourth-round clash with Switzerland's Stanislas Wawrinka, who defeated Marcos Baghdatis of Cyprus 6-3 6-2 6-7 (1) 7-6 (7).
Djokovic has yet to lose a set in the tournament and the 26-year-old was never in any danger against the 95th-ranked Sousa.
The 2011 U.S. Open champion faced just two break points on his own serve as he raced to victory in 100 minutes.
"Just doing my job, win in straight sets and spend as less time on court as possible," he said.
The Serbian's next opponent is Marcel Granollers, one of four Spaniards in the men's last 16, who knocked the last American out when he edged wildcard Tim Smyczek 6-4 4-6 0-6 6-3 7-5.
Bob and Mike Bryan did give America's men some cheer when they stayed on course to achieve a rare calendar-year grand slam in doubles after coming back to beat Canadians Daniel Nestor and Vasek Pospisil 6-7 (1) 7-5 6-2.
The Bryan brothers have already won each of the last four grand slam doubles titles, starting with last year's U.S. Open, but are bidding to claim all four in the same calendar year.
The only men to have achieved the feat were the Australian pair of Ken McGregor and Frank Sedgman in 1951. (Editing by Gene Cherry / Ian Ransom / Larry Fine)