Last year's runner-up, Sara Errani, also reached the last eight in straight sets.
Flipkens converted all four of her break points against Meusburger, and the third-seeded Errani broke Karin Knapp's serve five times to win their all-Italian match 6-4, 6-3.
Petkovic next faces Frenchwoman Alize Cornet, who needed three hours to beat Czech lucky loser Barbora Zahlavova Strycova 6-3, 6-7 (2), 6-3 in a match featuring 10 breaks of serve.
Zahlavova Strycova replaced second-seeded Petra Kvitova of the Czech Republic after she pulled out because of a sore throat.
"I really wanted to play, but I had to see the doctor because I wasn't feeling well," Kvitova said. "When I was trying to warm up this morning, after 10 minutes I realized that I couldn't breathe, so unfortunately I have to pull out."