Roger Federer’s conqueror had to contend with dropping his serve early in each of the first three sets to the Spanish 22nd seed before coming through in 3hr 15min.
“It all feels like a fairytale almost,” he said after holding his head in his hands in disbelief and sinking to the floor on Rod Laver Arena.
“I’m just living the dream, living what I’ve been working hard for,” Tsitsipas added.
“I feel a bit emotional but not too much -– I know I really worked hard to get here, playing in semis of a Grand Slam.
“I was asked my goals this year and said semis Grand Slam. And when I was answering this question, I thought I was crazy. But it is real. It just happened.”
Tsitsipas will face either world number two Rafael Nadal or unseeded Frances Tiafoe for a place in the final.
Federer did not win any of 12 break points in a four-set defeat against 14th seed Tsitsipas in round four, but Bautista Agut managed it on his first in the opening game of the match.
There was no panic from the 20-year-old from Athens and he broke back before stealing the set 7-5 with a second break created by a rasping forehand winner that flat-footed the Spaniard.
Bautista Agut went 2-1 up in the second set with another early break as the Tsitsipas first serve momentarily deserted him.
This time the Spaniard stayed on top and levelled the match 6-4.
Another break for Bautista Agut came in the fifth game of the third set.
But with the crowd beginning to get more involved, Tsitsipas outlasted the Spaniard in a gruelling game to level at 4-4.
And when another break followed the Athenian artist had his nose back in front.
By now his serve was beginning to flow as smoothly as his mane of long hair and there was no early break for Bautista Agut in the fourth set.
The gutsy Spaniard successfully served to stay in the match at 4-5.
But he was under pressure again at 5-6 and showed incredible resilience to repel a first match point after a 21-shot baseline rally and take it to a tiebreak.
But Tsitsipas jumped out to a 3-0 lead and secured the match on the first of four more match points.