Williams, the women's No. 1, reached her first Grand Slam quarter-final of the year by defeating Kaia Kenepi 6-3, 6-3, keeping alive her hopes for a third straight U.S. Open title.
Williams won 12 straight points on her serve to start the match against the 50th-ranked Kanepi.
The American said she felt pressure to win in New York after losing in the fourth round of the Australian Open, the second round of the French Open and third round of Wimbledon.
"I finally made a quarterfinal this year!" Williams said in her on-court interview, raising her hands to the cheers of the crowd in Arthur Ashe Stadium. "I think I felt it in my service game. I'm like, 'Can I please make it to a Grand Slam quarters this year?' So glad to do it in New York."
The 17-time Grand Slam champion next plays 11th-seeded Flavia Pennetta, a 7-5, 6-2 winner over 29th-seeded Casey Dellacqua.
Canada's Eugenie Bouchard, the seventh seed, is set to take on No. 17 Ekaterina Makarova later.
Bouchard, Williams and No. 10 Caroline Wozniacki are the only top-10 women's seeds to survive into the second week of the tournament.
Playing mostly mistake-free tennis, men's No. 1 Djokovic reached the quarter-finals in New York for the eighth year in a row — and 22nd consecutive Grand Slam tournament overall.
Djokovic, the 2011 champion at Flushing Meadows, made only 19 unforced errors, saved both break points he faced, and breezed past 22nd-seeded Philipp Kohlschreiber of Germany 6-1, 7-5, 6-4.
The last time Djokovic was eliminated before the quarterfinals at a major was a third-round loss to Kohlschreiber at the 2009 French Open.
Next up for the Serb, who reached the U.S. Open each of the past four years, will be a match against 2012 U.S. Open champion Andy Murray or ninth-seeded Jo-Wilfried Tsonga.
Fifth-seeded Canadian Milos Raonic plays No. 10 Kei Nishikori at night.
Third-seeded Stan Wawrinka faces 16th-seeded Tommy Robredo.
Canada's Pospisil eliminated in doubles
Canadian Vasek Pospisil and his American teammate Jack Sock were eliminated in the third round of the men's doubles event, losing 6-2, 6-2 to Argentina's Carlos Berlocq and Leonardo Mayer.
Pospisil and Sock were the surprise champions at Wimbledon in July.