The substance, commonly marketed as Mildronats, was banned on January 1 by the World Anti-Doping Agency. Sharapova claims ignorance over the ban.
"I don't want to end my career this way. I really hope that I'll be given another chance to play this game," said Sharapova in a press conference.
The Russian tennis star said she had used Mildronats for ten years, having first been advised to take the drug by a family-approved doctor in 2006.
Sharapova had been informed of the positive test on 2 March and she will be provisionally suspended from 12 March.
Several other athletes, like Russian Olympic gold medalist figure skater Ekaterina Bobrova, have admitted to testing positive for the drug.
Developed in Latvia, meldonium has been used to treat ischemia, or lack of blood flow. Meldonium had previously been on the World Anti-Doping Agency list of drugs to be monitored.
Mildronats is one of the most successful medicinal exports from Latvia. In 2013, export turnover of the drug reached €65 million. It is produced by the Latvian pharmaceutical company Grindeks.