Later, top seed Caroline Wozniacki won for the first time in over a month. The world's No. 1 player easily dispatched Polona Hercog of Slovenia 6-3, 6-0 as she began her quest for a fourth consecutive title here.
Vesnina was leading 2-0 in the first set and facing two break points when the quake, which was centered in Virginia, sent the Connecticut Tennis Center swaying, forced the evacuation of the stadium and caused a delay of over two hours.
Parts of the White House, Capitol and Pentagon were also evacuated after the magnitude-5.8 quake, which was felt all along the East Coast.
After play resumed, Vesnina held on to win the set, but she dropped the second and was tied at 4-4 in the third before rallying and breaking seventh-seeded Jankovic's serve in the final game of the match.
"I will remember this day for the rest of my life," Vesnina said. "There is nothing you can say. It's just like you won the match after the earthquake and it just feels great."
Jankovic arrived at the tournament on Monday after playing in the finals of last week's tournament in Cincinnati.
She said she felt flat and tired, especially after waiting for two hours in the grass outside the stadium while it was being cleared by the fire marshal and building inspectors.
"It doesn't make you feel good before going on the court," she said. "It's not like you are sitting on a couch. You are just sitting in the middle of nowhere and mosquitoes biting you and bugs and spiders crawling over you."
Wozniacki, who lost in her first matches at both Toronto and Cincinnati, had an easier time Tuesday. Playing with her boyfriend, U.S. Open golf champion Rory McIlroy, watching from the stands, Wozniacki broke Hercog to go up 5-3 in the first set and didn't lose another game.
"I'm definitely a player who likes to play matches and always plays better when I've gotten a few matches under my belt," she said. "So it definitely is nice that I could come back here and win my first match."
Wozniacki, who has never lost a match in New Haven, will face Christina McHale in the quarterfinals, a week after being beaten by the 19-year-old in their only other meeting.
The only American in the New Haven draw needed just 1 hour, 11 minutes to beat Carla Suarez Navarro of Spain 6-2, 6-2 on Tuesday.
But McHale said she doesn't view herself as a favorite against Wozniacki.
"She's the No. 1 player," she said. "She's a very good player and I'm not going to be any more confident because I beat her once."
Fifth-seeded Agnieszka Radwanska was also upset Tuesday, losing to Petra Cetkovska of the Czech Republic in three sets, 6-4, 5-7, 6-3.
Cetkovska was leading 3-1 in the second set before Radwanska rallied but was able to regain the momentum in the third set, breaking Radwanska in the final game.
"It was pretty difficult in my head to keep going because I was pretty tired and a little bit frustrated with myself," Cetkovska said. "But I kept fighting and I was trying to do what was working in the first set."