Down 2-4 in the final set, Angelique Kerber fought past Pauline Parmentier, 7-5, 3-6, 7-5, to reach the Indian Wells fourth round.
Seven-time Grand Slam champion Venus Williams booked a fourth-round berth with a 6-4, 6-2 victory over Lucie Safarova.
In the women's draw Angelique Kerber, who will reclaim the world No 1 ranking on Monday, breezed to a 6-2, 6-1 second-round victory over fellow German Andrea Petkovic at the BNP Paribas Open on Saturday.
Williams battled back from a break down in the final set to beat Chinese qualifier Peng Shuai 3-6, 6-1, 6-3.
Williams, seeded 12th in her second tournament since a runner-up finish to her sister Serena in the Australian Open in January, had saved three match points en route to a second-round victory over Jelena Jankovic.
After an exchange of service holds to start the final set with, it seemed as though we had a victor for the day as Babos made the first breakthrough and broke serve first, taking an early 2-1 lead as she had already won eight of the last ten games. "I'd had great practices in the last few months so I'm ready to stay on court for three hours".
The ultra-fit Kerber persevered despite seeing her opponent save 13 break points in the match.
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After the Hungarian managed to consolidate the break with a tough hold of serve, Vesnina finally got onto the scoreboard in the second set as she had a nervy service hold, lessening the deficit to just two games.
"If things get closer, then I think my better tennis is going to come". But in her Indian Wells opener on Saturday, she appeared to rediscover the form that led her to two Grand Slam titles a year ago.
"It was not so easy", said Kerber after the match.
"I have fallen in love with the game again", she said of her resurgence.
"When you are in a match like this, you are not thinking about pressure", Kerber said. "The balls are flying a little bit different, as well". "So she took the game in her hands".
Elsewhere, No. 3 seed Karolina Pliskova advanced when 15th-seeded Timea Bacsinszky retired trailing 5-1 in the first set and French Open champion Garbine Muguruza of Spain defeated No.10 seed Elina Svitolina 7-6 (7-5) 1-6 6-0.
Eighth-seeded Svetlana Kuznetsova, 31, was the first to advance with a 6-1 6-4 win over Caroline Garcia of France.