* Serena outguns Stephens in all-American battle
* Murray overcomes steamy conditions for straight-sets win
* Former champion Hewitt continues his run (Updates with Serena Williams winning)
NEW YORK, Sept 1 (Reuters) - Serena Williams avenged her loss to Sloane Stephens earlier this year by thrashing her fellow American in straight sets on Sunday to reach the quarter-finals of the U.S. Open.
The world number one showed no mercy against Stephens, crushing the rising star 6-4 6-1 to remain on course to defend the U.S. Open title she won last year and restore the sport's world order.
"How excited are we about the future of American tennis?" Williams asked in a courtside interview. "It definitely felt like something bigger."
Stephens beat Williams in January in the quarter-finals of the Australian Open to enhance her reputation as one of the leaders of the next generation but was outgunned on Sunday.
The 31-year-old Williams, bidding to become the oldest woman to win the U.S. Open since tennis turned professional in 1968, won a tight first set then ran away with the second.
"Obviously she's number one in the world for a reason," Stephens said. "I thought she played really well.
"Obviously it didn't go how I wanted. The second set got away from me a little bit. All in all I thought I competed well and played well. That's all you can do really."
Williams advanced to play Carla Suarez Navarro in the quarters after the Spaniard upset Germany's eighth-seed Angelique Kerber 4-6 6-3 7-6(3).
The defending men's champion, Andy Murray, was also in a ruthless mood against Germany's Florian Mayer, winning their third round match 7-6(2) 6-2 6-2 in less than two hours.
Murray struggled to impose himself on Mayer in the opening set as both players tried to adapt to the gruelling conditions at the Arthur Ashe Stadium center court.
It was hot and steamy and the wind was gusting but once Murray found his rhythm it was all smooth sailing. The defending champion won the first set tiebreaker then broke Mayer's serve four times to close it out in straight sets.
"It was tough conditions, not tricky. 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).
"Today was good," Berdych said. "It was a bit hot, a bit humid, but otherwise tennis was good.
"Finally, I managed to stay focused from the first point till the last. I didn't do any ups and downs, just took all my chances that I had and won in straight sets."
Australian Lleyton Hewitt continued his run through the draw when he beat Russia's Evgeny Donskoy 6-3 7-6(5) 3-6 6-1.
Hewitt won the U.S. Open 12 years ago but the 32-year-old has turned back the clock this week to reach the last 16 for the first time since 2006.
Bob and Mike Bryan's bid to achieve a rare calendar-year grand slam in the men's doubles remained intact after they survived a tough center court clash.
The American twins switched sides after dropping the first set then found themselves down a break in the second before rallying to beat Canadians Daniel Nestor and Vasek Pospisil 6-7(1) 7-5 6-2.
"It ended up working out but it was just a desperate call," said Bob Bryan.
"We were feeling a little bit hopeless on the return games, and throwing in a switch like that sometimes is a psychological advantage."
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)