But top seed Angelique Kerber of Germany advanced with a 7-6 (7/3), 6-2 win over China's Duan Ying-Ying despite having to wait over six hours to play due to a schedule back-up caused by rain delays.
Vesnina, seeded 13th, faced the rain delays and wind during her closely fought contest but ended up losing 3-6, 6-4, 7-5 to 594th-ranked Tomljanovic, who will face Czech Lucie Safarova in the third round.
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Kerber's game was scheduled for late afternoon on center court but the world number one had to wait until around 11:30 p.m. for a match moved to one of the outside courts.
"It was tough because it was really windy, the conditions are really tough when you play so late, I was really trying to stay positive and not think too much of the conditions," Kerber told AFP, adding that she was unhappy with the decision to move her off center court.
"I don't think it was fair but what can I do? It was not my decision but I think it is not really fair to me," she said.
While Kerber was bumped off the main stadium, a men's contest between ninth-seeded Bulgarian Grigor Dimitrov and lowly ranked Argentine Guido Pella took place on the prime court before local favorite Venus Williams followed.
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Williams had little trouble defeating Brazilian wildcard Beatriz Haddad Maia 6-4 6-3.
Williams, whose sister Serena is missing the tournament through injury, will face Romanian Patricia Maria Tig in the next round.
American qualifier Taylor Townsend upset Italian 25th seed Roberta Vinci 6-3, 6-2.
The 20-year-old left-hander's victory over the 2015 US Open runner-up was among her top career triumphs and sets up a meeting with seventh seed Svetlana Kuznetsova of Russia, who downed Mandy Minella of Luxembourg 6-2, 6-2.
"I played really well and I am happy with that result," Townsend said of her victory over Vinci. "I've had some tough matches here. I think I was a lot more solid than I have been in the last couple of days. "All the things that didn't really work... it came together today."
Townsend has never played Kuznetsova, the 2004 US Open and 2009 French Open champion, and acknowledged it would be a huge challenge.
"I really like watching her play," Townsend said of Kuznetsova, runner-up at Indian Wells last week. "She has won two Slams and is a great player who is playing some of her best tennis now. I am playing well. I've got four matches under my belt and I don't have anything to lose."
Spanish sixth seed Garbine Muguruza had all night to ponder a possible early exit as she trailed 6-0, 3-2 to Christina McHale when play was suspended due to rain on Thursday night. But Muguruza, the reigning French Open champion pulled out the stops, saving a match point to earn a 0-6, 7-6 (8/6), 6-4 win.
She saved a match point at 6-5 in the second set tie-break and trailed by a break to begin the third set before turning the match around.
"I think I had a second chance with the rain," said the Venezuelan-born Muguruza. "I was missing by a little and she was not missing at all. I just forgot about it, and I woke up today and I said, 'Hey, this is not over.' I went to the court, I fought and I got it in the end."
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