Tsitsipas moved into his first grand slam semi-final after a tough 7-5 4-6 6-4 7-6 (7-2) victory over the Spanish 22nd seed on Rod Laver Arena.
The 20 year-old, who is the 14th seed, managed to back up his stunning win against Roger Federer with another success as Bautista Agut – coming off three five-set wins in four matches – tired late.
Awaiting Tsitsipas in the last four is either Rafael Nadal or Frances Tiafoe.
Tsitsipas took time to settle, falling behind an early break, but he recovered late in the first set.
As Bautista Agut struggled with the sun at one end, Tsitsipas created set point in the 10th game with a forehand winner and converted when the Spaniard sent a backhand into the net.
A poor game saw Tsitsipas give up serve in the third game of the second set and this time Bautista Agut made no mistake on his way to levelling the match.
For the third straight set, Bautista Agut went 4-2 up, only to begin to tire, losing four consecutive games.
Tsitsipas, who gave up multiple first serves due to time violations, took the set 6-4 after a slice backhand down the line.
Both players were comfortable on serve in the fourth set until the 12th game, when Bautista Agut saved a match point with an incredible forehand winner on his way to forcing a tie-break.
Tsitsipas took a 3-0 lead courtesy of a stunning backhand down the line and while Bautista Agut briefly responded, he closed out his victory.