The fifth-seeded 29-year-old earned an early first-set break before holding off a Vesely fightback to boss the tiebreak and win the set. Out of sudden, Juan Martin created the only break point of the set and he defended his backhand nicely in the following point to get an error from Bautista Agut, clinching the set by 6-4 and sending the match into a decider.
However, Del Potro broke at the ideal time - in the 10th game - in the second set to level the match.
The ex-US Open victor and 2009 Classic champion was broken in each of his three first-set service games and didn't rebound until late in the second.
"Today was a very tough match, I don't know how my body will be tomorrow, but now I want to enjoy the win and we'll see", Bautista Agut said.
"Juan Martin is very tough to beat and I think I just played two unbelievable games at the end of the match".
Del Potro beat Russian Karen Khachanov 7-6 (7/4), 6-3 in his quarter-final. Del Potro served better and better as the match progressed and he dropped just 3 points in 5 service games, with another solid hold presumably coming up in game 11. I've been fighting a lot on the court...
"I started the match very well".
Earlier, fifth seed Bautista Agut outlasted Dutchman Robin Haase in an epic clash that took nearly three hours before the Spaniard produced an audacious backhand lob that kissed the baseline to clinch the 6-7 (7) 7-6 (3) 7-6 (5) win. "But then my game improved a little bit and I made mistakes in the last game", del Potro said. I wanted to play very aggressive.