In a match that ended at 1:46 a.m. local time (0646 BST), the 15th-seeded American recovered from a woeful start to down 17th seed Vesnina and set up a meeting with Elina Svitolina.
Svitolina downed local favourite Shelby Rogers 6-4 7-5 but there will still be five Americans in the women's last 16, with Venus Williams, Sloane Stephens, Jennifer Brady and CoCo Vandeweghe also through.
Brady battled to a 6-3 4-6 7-6(3) defeat of Monica Niculescu, while Vandeweghe beat 10th seed Agnieszka Radwanska 7-5 4-6 6-4.
Keys shrieked a long 'come on!' when Vesnina buried a shot into the net on the second match point.
"You guys were amazing, I definitely did not show up very well for the first set but you helped me get back into the match, thank you for helping me through that one," Keys told the crowd.
"I think I finally settled in after being a bit over-amped at the start. Who doesn't love a late show?"
(Reporting by Julien Pretot; Editing by Ian Ransom)
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