The 30-year-old Russian, three-time 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 in front of a supportive crowd 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.
REDUCED BAN
Sharapova's initial two-year suspension was reduced to 15 months after she tested positive at the 2016 Australian Open for meldonium, a medication the Russian had been taking within the rules but which was then reclassified as a banned drug.
"It's important to play, points, games, sets. It is a journey that officially starts today and I look forward to playing as many matches as I can," said Sharapova, the second highest paid female athlete last year according to Forbes.
"I spent a long time without hitting any balls. I didn't know when I would be back. I went to school for a little bit, I grew my business and had a normal life. I put the racquet away for a little bit.
"I felt I had to grow as a person and I felt I had to step up and do it."
Sharapova was clearly nervous at the start, firing three forehands long in the first game, double-faulting on her first service game point and being broken by the Italian who took a quick 2-0 lead.
It took 15 minutes for Sharapova to win her first game but she gradually improved her service, started attacking Vinci's and clinched the first set after an hour.
Sharapova, growing in confidence with every point despite a dozen unforced errors in the first set, broke the Italian early in the second, and her 11th ace put her 5-3 ahead.
She broke the 34-year-old again to secure her first win on her first match point.
(Reporting by Karolos Grohmann,; Editing by Neville Dalton and Toby Davis)
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