The 22nd-seeded Spaniard captured the decisive break in the ninth game of the final set before serving out strongly to claim a 6-7(6) 6-3 6-2 4-6 6-4 win at Margaret Court Arena.
Bautista Agut, who had lost the fourth-round match of a Grand Slam in each of his nine previous attempts, threw his arms up in the air and roared in triumph.
It was Bautista Agut's third five-set win this week at Melbourne Park, continuing a remarkable run that has also included victories over an injury-hobbled Andy Murray, Australian John Millman and 10th seed Karen Khachanov.
"I've been playing very good tennis," the 30-year-old said.
"I did a great job this pre-season and it's been a great month for me and for my tennis."
Bautista Agut will next face 20-year-old Greek Stefanos Tsitsipas, who overcame defending champion Roger Federer in one of the biggest upsets of the tournament this year.
"Now I want to recover well, to enjoy the win I had today. We will see on Tuesday," he added.
In a splendid start to the season, Bautista Agut claimed his ninth career title at the Qatar Open earlier this month, beating Stan Wawrinka, Novak Djokovic and Tomas Berdych in consecutive matches to lift the trophy.
(Additional reporting by Hardik Vyas; Editing by Clare Fallon)
