The 10th seeded Italian was joined by 14th seed Sloane Stephens of the United States, who overcame Kristina Mladenovic of France 6-2 7-6(5).
Vinci got off to a slow and struggled against the German's big forehand in the opening set. However, she mixed up her attack in the second and third sets to upset Goerges' rhythm and came out on top.
"In the third I tried to play aggressive and to try to put a lot of balls in the court because sometimes she plays an unbelievable game, and then after not so good," Vinci said.
"So she has a lot of ups and downs. I didn't play good tennis today. But important thing was I won."
World number 12 Vinci, who has struggled on faster surfaces this season but won claycourt titles in Katowice and Palermo, said the conditions and Goerges' unpredictability made it a difficult match.
"For me, it was tough," Vinci said. "Especially because there was a lot of wind and it's not easy to play confident.
"But in the end she played so bad, and probably she's not confident and that's why I won today."
American Stephens, who was coming of an opening round upset in the first round in Washington last week, got into her rhythm quickly against the big-hitting Mladenovic but after winning the first set she had trouble closing the match out.
"She upped her game a lot in the second set, and sometimes it's going to be a little up and down," the 20-year-old said.
"But I thought I stayed pretty solid and got through it. I knew she's probably going to press a little bit and make me play more balls. I just tried to stay with it and hit a lot of angles and move her a bit. It seemed to work and I played solid so it was good."
Britain's Laura Robson withdrew prior to her singles match against Yanina Wickmayer with a right wrist injury and was replaced by lucky loser Bethanie Mattek-Sands, who was beaten 6-4 2-6 6-4 by the Belgian.
Serena Williams is the tournament's top seed. World number two Victoria Azarenka and number three Maria Sharapova both withdrew from the event with injuries.
(Editing by Peter Rutherford)
