Indian Wells champion Juan Martin del Potro of Argentina, the fifth seed, was more fortunate, winning his 13th consecutive match by defeating Japan's Kei Nishikori 6-2 6-2.
The big-hitting Cilic fired six aces and saved all three break points he faced against Pospisil, but the 77th-ranked Canadian made it more difficult than the Croat probably expected.
Trailing 5-6 0-40 in the second set, the Canadian saved three consecutive match points and hit a running forehand pass to cap a 33-shot rally before Cilic claimed victory in the tiebreak.
"It was a relief at the end," Cilic told reporters. "I had those three match points at 6-5 and there was that great rally at deuce. I was playing really well in the tie-break, returning really well and serving really good.
"I saved the best for last and I'm really pleased with the victory."
Del Potro fired 17 winners, claiming all five points when crashing the net and turned back Nishikori's lone break chance to take the match in 72 minutes.
"I think I'm playing good," Del Potro told reporters. "I can play even better, but I need to get confidence playing on this court, trying to keep winning matches. But it is not easy. My body is starting to feel tight because of all my success."
The fifth seeded Argentine will face Filip Krajinovic in the fourth round on Tuesday after the Serbian defeated Frenchman Benoit Paire 6-3 6-3.
Meanwhile, South Korean Chung Hyeon, the 19th seed, needed just 62 minutes to dispatch American qualifier Michael Mmoh 6-1 6-1.
The highest-ranked player from Asia at number 23, Chung has not dropped a set in the tournament and will take on Joao Sousa in the last 16 after the Portuguese outlasted American Jared Donaldson 1-6 6-3 6-4.
Elsewhere, big-serving Canadian Milos Raonic pounded 10 aces to take down Argentina's Diego Schwartzman 7-6(5) 6-3.
(Reporting by Gene Cherry in Salvo, North Carolina; Editing by Ken Ferris)
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