Cilic was already on borrowed time having survived two match points to beat Bautista Agut’s compatriot, Fernando Verdasco, in the third round.
And after storming back to take the match into a deciding set on Sunday, the Croat’s big-match experience looked to have him on the front foot.
Cilic wasn’t without his chances, as break points went begging early in the fifth, but it was the Spaniard who finally landed the critical blow at 4-all.
He broke on his third opportunity, in a near seven-minute game, as he ripped a forehand down the line.
And there were no sign of nerves as he closed in on a long-awaited career first.
Bautista Agut brought up two match points with a crosscourt forehand winner and claimed it when his opponent’s backhand found the net.