The 22nd-seeded Bautista Agut set up match point with a forehand that nicked the side baseline, then clinched it when sixth-seeded Cilic hit a weak backhand that didn't reach the net.
Bautista Agut's win in 3 hours, 58 minutes advanced the Spanish player to a quarterfinal against either six-time champion Roger Federer or 20-year-old Stefanos Tsitsipas.
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