"I think I played a really good match," the Russian said in her courtside interview.
Vesnina blasted 28 winners to 21 unforced errors and broke Kerber's serve five times to forge ahead before holding off a late rally by the German, who will return to number one in the world rankings next week despite her early exit in California.
"I was a little nervous at the end of the match," she said about Kerber winning eight of nine points to recover from 4-1 to 4-3 in the second set. "So I'm really happy that I closed that match."
Seven-times grand slam winner Williams, the 12th seed, recovered from a slow start to grind down China's Peng Shuai.
After battling back from match-point down earlier in the week against Jelena Jankovic, Williams executed another turnaround against Peng to win 3-6 6-1 6-3.
"There were some up-and-downs and errors," Williams told reporters. "But I feel like I got my focus more in that second set and towards the end of the third."
Vesnina leads their head-to-head 3-2 and won their most recent match at last year's Miami Open.
World number three Karolina Pliskova advanced to the last eight after 15th seed Timea Bacsinszky of Switzerland retired from their match while trailing 5-1 in the first set.
The Czech will face reigning French Open champion Garbiñe Muguruza of Spain, a 7-6(5) 1-6 6-0 winner over in-form Ukrainian Elina Svitolina.
Caroline Wozniacki, the 13th seed and 2011 champion, reached the quarters with a 6-4 6-4 victory over ninth-seeded American Madison Keys. The Dane will meet Kristina Mladenovic of France, who beat American Lauren Davis 6-3 6-3.
Eighth-seeded Svetlana Kuznetsova, 31, was the first to advance with a 6-1 6-4 win over Caroline Garcia of France.
Kuznetsova has reached the quarters in six of her last eight tournaments, dating back to Tianjin in October 2016.
"I have fallen in love with the game again," she said of her resurgence.
Kuznetsova will compete in an all-Russian quarter-final against Anastasia Pavlyuchenkova, a 6-4 3-6 6-2 upset winner over fifth seed Dominika Cibulkova of Slovakia.
(Reporting by Larry Fine in New York; Editing by John O'Brien)
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