Williams defeated US 15th seed Sloane Stephens 6-4, 6-1 to reach the US Open quarter-finals, winning eight of the last nine game in avenging a loss that ousted her from the 2013 Australian Open.
"It definitely felt like a bigger match because Sloane is such a great player, but I had to stay focused in the moment," Williams said. "I just tried the whole time to do what I wanted to, stay calm and stay relaxed."
Also booking a quarter-final matchup were Chinese fifth seed Li, who beat Serbian ninth seed Jelena Jankovic 6-3, 6-0, and Russian 24th seed Ekaterina Makarova, who upset Polish third seed Agnieszka Radwanska 6-4, 6-4.
Li, the 2011 French Open champion, matched her best US Open result by reaching the last eight.
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"Maybe today was the best match of my whole life," Li said, adding her strategy against Makarova would be to "Play aggressive. I wish I can play the same level like today."
"Last time I lost to her in Rome I was feeling so sad. I was feeling I played so good. I said, 'You have to remember how you felt after Rome and not have that feeling any more.'"
Suarez Navarro hit 45 winners and took advantage of 50 unforced errors by Kerber to rally for the victory after two hours and 41 minutes. Radwanska dropped the last eight games and was ousted in the fourth round at the US Open for the fourth time, having never reached the quarter-finals.