It is the second time Bautista Agut has beaten Djokovic in his career and both victories occurred in a semifinal.
"It's been a fantastic week for me", Bautista Agut, who endured some nervy moments in the decider, said. We don't know the draw yet, I don't know who my first-round opponent is, I don't know if I play on Monday, if I play day or night. "I can still get to one or two levels up, but I thought under the circumstances I managed to find a way to win against a player that is hitting the ball very fast and just being very aggressive", he added. That's why Novak is No. 1 in the world. I'm happy with the way I'm playing.
Bautista-Agut's first round match was scheduled for Tuesday, which would have given him little time to acclimatize, and also recover from the long 17-hour journey from the Middle East.
Djokovic lost two of his first three service games against Basilashvili and dropped the first set, as he had done in his last-16 win over Marton Fucsovics on Wednesday.
Defeat saw Djokovic's 13-game winning run in Qatar come to an end, leaving the player frustrated and smashing his a racket in the second set.
Herbert, who beat second seed and World No.8 Dominic Thiem 24 hours earlier, will now play Tomas Berdych in the last eight, who won a three-set match against Fernando Verdasco, 4-6, 6-3, 7-5, in the last match of the night.
After securing his semi-final berth, the irrepressible Djokovic then headed back out to court to play his sixth match of the week in the doubles with his brother Marko. "Especially in the third set he played well".
It is the first time Djokovic has won back-to-back matches from a set down since Cincinnati in August previous year, a tournament he went on to win.