Most of the opening-round matches scheduled Sunday were postponed because of the rain that persisted throughout the day.
Murray, however, had the benefit of playing under the roof of Wimbledon's Center Court, as did other winners Novak Djokovic and Maria Sharapova.
Murray, who gave British tennis fans a home-grown finalist at the Wimbledon Championship earlier this month before he lost to Roger Ferderer, had no trouble disposing of Stanislas Wawrinka of Switzerland 6-3, 6-3.
Djokovic, who was a bronze medal winner for Serbia four years ago at the Beijing Olympics, rallied to defeat Italy's Fabio Fognini 6-7 (7-9), 6-2, 6-2.
Sharapova, of Russia, seeded third in the women's draw, recorded a 6-2, 6-0 win over Shahar Peer of Israel.
Fourth-seeded David Ferrer of Spain was one of the few others who was able to complete his match, defeating Vasek Pospisil of Canada 6-4, 6-4. No. 5 seed Jo-Wilfried Tsonga of France survived against Thomaz Bellucci of Brazil 6-7 (5-7), 6-4, 6-4.