Playing under Wimbledon's retractable roof on Centre Court because of rain, No. 3 Sharapova served well, returned aggressively and swept the final eight games.
Sharapova completed a career Grand Slam last month when she won the French Open. She carried the Russian flag in the opening ceremony after missing the 2008 Olympics because of a shoulder injury.
Sharapova improved to 7-0 against Peer.
While the roof was closed from the start of the day, 22 matches on other courts were postponed because of rain, including one involving top-seeded Victoria Azarenka of Belarus.