Former champion Murray, 27, angrily smashed his racket after double-faulting to fall 5-3 down but Pospisil could not convert his one set point when he served at 5-4.
Murray will play Frenchman Gilles Simon, the eighth seed, next after he beat compatriot Jeremy Chardy.
"Gilles is a tough opponent. He's a strong runner with perfect timing," Murray said. "I'll need to serve better than today."
Defending champion Tomas Berdych also reached the quarter-finals with a 6-0 3-6 6-3 defeat of Italian Andreas Seppi.
(Reporting by Martyn Herman, editing by Pritha Sarkar)
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