The three-time Grand Slam champion eased to a 6-3, 6-3 victory, reaching his first ATP Tour quarterfinal this year.
Murray, 34, dominated the opening exchanges, nearly breaking Pospisil in the first game of the match before winning four straight games to claim the first set.
Pospisil looked to recover in the second, breaking Murray to begin the set before holding his serve to take an early 2-0 lead. But Murray rebounded, taking six of the next seven games to achieve a comfortable victory.
The result marked the first time that Murray won consecutive matches since his run to the third round at this year's Wimbledon Championships.
Murray beat France's Ugo Humbert on Tuesday to set up the tie against Pospisil, and he said after that he is confident that his game is on an upward trajectory and he can compete against top players once again.
"I don't think I've played that well recently, but my body has been good even though I'm not playing my best, I've created opportunities to win sets against a number of top players," Murray said.
"There is no question that I can play better. The thing that has been frustrating for me in some of the matches is that I had opportunities... If I take those chances, those matches are very different."
He had hip surgery in 2018 and 2019 and has made adjustments to his serve motion, but has not been to recover his previous world-class form.