Djere dominated the tiebreak but eventually needed five set points to close out the first set.
Spaniard Bautista Agut responded by cutting down unforced errors in his game before converting a decisive break in each of the next two sets to book a place in Sunday's final.
Earlier in the day, world number 84 Matteo Berrettini reached his first tour final with a 6-4 7-6(6) win over Estonian Jurgen Zopp.
The 22-year-old Italian has played some of the best tennis of his young career this week, ousting 2016 champion Feliciano Lopez and fourth seed Andrey Rublev.
Persistent showers delayed the start of the semi-finals by more than three hours, but Berrettini looked in complete control as he saved both break points against Zopp, and struck nine aces to book a spot in Sunday's final.
"The conditions were different today after the rain," said Berrettini, who is yet to drop a set in the tournament.
"I managed to raise my level and was fighting hard during the whole match.
"I've been playing a lot of important matches lately - winning a round in Rome, my first match at a slam (at Wimbledon) and more. I'm feeling well and excited about the final tomorrow."
(Reporting by Hardik Vyas in Bengaluru, editing by Pritha Sarkar)
