The 30-year-old Russian, a three-times winner on Stuttgart's clay courts, received a controversial wild card for the German tournament, having had no ranking points after more than a year out following her suspension.
She had a nervous start but quickly found her strokes and her trademark shrieks to power past the world number 36 7-5 6-3 and set up a second-round clash against fellow Russian Ekaterina Makarova.
"It's the best feeling in the world. To know I would be walking back into the arena was very special," Sharapova said in a courtside interview.
"I was waiting for this moment for a long time." She added: "When I am on the court, because I have been doing it for so long, even though you are rusty and trying to get a rhythm, you try to block everything out. I am a competitor by nature. That's when I am at my best."
Sharapova has also received invitations to play in Madrid and Rome and will find out in May whether she will be given a wild card for the French Open.
Some players, including Vinci, have criticised the wild card awards, saying a doping offender should have to start from scratch and build up their ranking by playing in the lower tournaments again.
"I am not an individual that's angry or bitter. I let things go," she told a post-match news conference. "I am being offered wild cards from tournament directors. I am not getting a wild card to receive a trophy or a gold platter.
"I still have to go through matches and win them."
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