It took sixth-seeded Italian Vinci, 33, just over two hours to get past the 28th-ranked Hungarian 6-3, 3-6, 6-4 in a replay of last year's quarterfinal.
"It's always tough to play against Timea," Vinci said. "She's a great player with a very good serve.
"Today it was really difficult to change the course of the match. And I'm really happy to win because it's always hard to play in the first round.
"But I love this tournament as I have great memories from the last year."
Vinci looked in command, winning the opening set in 37 minutes, but Babos struck back in the second and appeared to take command when she broke at the start of the third.
