Kanepi won just six points in the opening set as Williams laid down a marker, but she fought back and used her own ferocious power to blast through the six-time champion.
"Winning a big game and a very important game and a really tight game, I think it was just a relief", said Williams, who took control with a break in the next game and maintained that advantage until she fired a forehand victor on her first match point.
"I know she was the best at that time, but I just wanted to win".
Overall, the 23-time major champion struck 46 winners - with 18 aces - to advance to the last eight of the US Open for the 15th time in her career.
"I think just like I always say, get your racquet on it, make a play on the ball, make your opponent play an extra ball", Stephens told reporters. "That was the most important thing for me to improve, being fit and able to play long matches". "I worked really hard on that".
Tottenham star Son Heung-min spared military service after Asian Games win
On September 1, the South Korean soccer team won the final match after scoring 2-1 against Japan at the '2018 Asian Games' . The Iranian side dominated the match from the very start and won 25-17, 25-22, 25-21, to successfully defend their title.
She dropped her first set of the tournament Sunday, two days after routing sister Venus 6-1, 6-2 in the third round.
Williams will move on to play No. 8 seed Karolina Pliskova in the quarterfinals.
Colin Kaepernick (center) and Eric Reid (right) attend the third round of the U.S. Open tennis tournament at the USTA Billie Jean King National Tennis Center on Friday in NY.
The importance of a hard-fought hold for Williams at the start of the third was highlighted by a loud cry from the American as she converted a backhand victor, which drew prolonged applause from the crowd.
It was the first set she had taken off the American in five meetings. That put her into the match against Kanepi, the 44th-ranked Estonian who upset top-ranked Simona Halep in the first round and is seeking her second consecutive quarterfinal in Flushing Meadows.
Tottenham's Heung-Min Son one game from military service exemption
Son played in Tottenham's season opener against Newcastle this month before joining up with his South Korean team-mates. South Korea's Son Heung-min celebrates scoring at the World Cup at Kazan Arena, Kazan, Russia - June 27, 2018.
If she defeats Pliskova she could face defending champion and fellow American Sloane Stephens in the last four. "I'm going to continue to grind and see what happens".
Asked if she was already thinking about a little brother or sister for Olympia, she said not quite yet. She missed her Australian Open title defense, not returning to the tour until March.
"Right now is not the time", she said. I'm feeling great now, too.
Sevastova toppled seventh-seeded Elina Svitolina 6-3, 1-6, 6-0, leaving just two of the top 10 women's seeds remaining.
Huawei unveils its new Kirin 980 mobile processor
Around front, the new handset has a 24MP "Fusion Pixel" camera with an aperture of f/2.0 that's also coupled with a 2MP camera. The Cortex A76 cores each feature 512KB of L2 Cache and the Cortex A55 cores feature 128KB of L2 Cache memory.