The fourth-seeded Murray is aiming to give Britain its first male champion at the All England Club since 1936. He lost in the semifinals in 2009, 2010 and 2011.
His match against Cilic on Court 1 was suspended on Monday in the second set because of showers, was delayed Tuesday because of more rain, then was interrupted again in the second set.
Murray ended Cilic's eight-match winning streak.
In the quarterfinals, Murray will face No. 7 David Ferrer, who eliminated No. 9 Juan Martin del Potro 6-3, 6-2, 6-3. Murray and Ferrer have split 10 previous head-to-head matchups.
