Van Uytvanck – ranked 91 places behind her German opponent in the world rankings – won the first set on a tie-break and had plenty of chances to seal a straight-sets win.
But Petkovic, seeded third, battled back to edge a second set tie-break then eased through the decider to clinch the second round match 6-7 (5/7) 7-6 (7/5) 6-2 in three hours and 19 minutes.
In earlier first-round action, more home hopes were dashed as Kirsten Flipkens went down 6-3 6-2 to Monica Niculescu of Romania, but Dutch qualifier Indy De Vroome surprised Tsvetana Pironkova 5-7 6-3 6-3.
Fifth-seeded Spaniard Carla Suarez Navarro survived a nightmare first set to see off Italy’s Camila Giorgi 1-6 6-2 6-4 and book her place in round two.
Other successful seeds included Slovakia’s Dominika Cibulkova, who beat Kateryna Bondarenko 6-1 6-2, and eighth seed Karolina Ploskova, who overcame veteran Mirjana Lucic-Baroni 6-4 7-6 (5).
Kristina Mladenovic advanced when Spanish opponent Silvia Soler-Espinosa retired when trailing 6-2 1-0.