The win moved Cilic - who won his ninth ATP Tour title two weeks ago in Zagreb, Croatia - into the quarter-finals against fifth-seeded Kei Nishikori.
Haas advanced to the second round with a 7-6 (6), 6-4 victory over Russia's Evgeny Donskoy on Wednesday.
Denis Istomin took advantage a sluggish John Isner to eliminate the third-seeded American 6-4, 6-3.
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