Kei Nishikori beat American Ryan Harrison 6-4, 6-4, while defending women's champion Flavia Pennetta easily advanced on a 91-degree day in the desert.
Murray broke Pospisil three times in the first set, winning the final six games. They traded breaks to open the second. Murray broke Pospisil at love in the eighth game and then served his own love game to close out the match.
Two-time winner Maria Sharapova was to play later, while defending men's champion Novak Djokovic competes in an evening match.
Murray lost in the Australian Open final to Djokovic and has played just two other tournaments so far in addition to Davis Cup.
Nishikori dispatched Harrison in 1 1/2 hours despite struggling with his first serve. The fifth-ranked Japanese player reached the third round at Indian Wells for the first time in eight tries.
Nishikori arrived in the desert on a roll, having reached the quarterfinals or better at each of his four previous tournaments this year. He won the title in Memphis.
Harrison fell to 1-24 against top-10 players in his career.
Pennetta beat American Madison Brengle 6-4, 6-2 as the Italian opened defence of the biggest title of her career. She has more ranking points (1,065) on the line at this tournament and coming up in Miami than any other player. Pennetta is looking to rebound from a poor start this year, having lost in the first round in Sydney and the Australian Open.
Other women's winners were: No. 4 seed Caroline Wozniacki, fifth-seeded Ana Ivanovic, No. 11 Sara Errani, and No. 16 Madison Keys.
On the men's side, Juan Monaco upset No. 10 Marin Cilic 6-4, 6-4, while No. 16 Kevin Anderson, No. 26 Phillip Kohlschreiber, and No. 28 Fernando Verdasco all won in straight sets.