World number 28 Vinci looked to be on her way out of the tournament when Wickmayer, ranked 30th, had match point on serve in the eighth game of the third set.
But a double fault opened the door for Vinci to break, and she never looked back as she stormed to a 6-2 3-6 7-5 triumph.
In the late match, Aleksandra Wozniak of Canada overcame a rain delay to oust Slovakia's Daniela Hantuchova 7-5 7-6 (7/5).
Another home favourite looked to be on her way to the second round when Stephanie Dubois breezed through the first set against Chanelle Scheepers, but the South African hit back to win 1-6 6-3 6-0.
Carla Suarez-Navarro prevailed 1-6 6-4 6-3 in the all-Spanish battle with Maria Jose Martinez Sanchez, while Kazakhstan's Sesil Karatantcheva stunned Sorana Cirstea of Romania 6-0 6-0.
There were also first-round wins on Tuesday for Russian pair Ekaterina Makarova and Ksenia Pervak, Austrian Tamira Paszek, Slovakia's Jana Cepelova and Romania's Simona Halep.