The Russian 30th seed, who was suspended for 15 months for taking banned drug meldonium in 2016, broke her 23-year-old opponent twice in the opening set and peppered the court with powerful groundstrokes in a superb display of hitting.
Sharapova, who beat Harriet Dart 6-0 6-0 in her opening round, won seven consecutive games to take a 4-0 lead in the second set before world number 64 Peterson could get on the board. The Russian did not face a single breakpoint in the match and sealed it in 71 minutes with a third break in the set.
In a clash of former world number one players, the five-times Grand Slam winner will next meet Dane Wozniacki, who eased past Johanna Larsson 6-1 6-3 earlier in the day. (Reporting by Sudipto Ganguly; editing by Christian Radnedge)