The in-form Spanish 22nd seed will play in his first Grand Slam quarter-final after winning 6-7 (6/8), 6-3, 6-2, 4-6, 6-4 in two minutes under four hours on Margaret Court Arena.
Bautista Agut began his season by beating Novak Djokovic en route to the Doha title and now faces either defending champion Roger Federer or Stefanos Tsitsipas for a place in the semi-finals.
He began his Australian Open campaign with an epic five-set win over Andy Murray, in what could be the three-time Grand Slam winner's last match because of a chronic hip injury.
His second round match against Australian John Millman also went the full distance before a straight-sets third round win against 10th-seeded Russian Karen Khachanov.
On Sunday he ended Croat Cilic's hopes of a repeat match-up against Federer in the quarter-finals after losing to the Swiss master in five sets in last year's final.