World number one Djokovic received a standing ovation from the home crowd by the Danube river in the Serbian capital after an erratic display against world number 255 Molcan, who battled through from the qualifying rounds.
Djokovic also lost to Russian Aslan Karatsev in the Serbia Open semi-finals before being defeated by Rafael Nadal in the Rome Masters final and his display against Molcan was at times unconvincing.
"Of course I'm disappointed not to win it, but at the same time I'm very pleased with the level of tennis that I managed to find in the later stages of this tournament.
"I have to obviously keep the momentum going I feel with the quality of tennis," he said. "Do more training, of course, and few more matches. I think I'm on the right way.
"(This is) my favourite tournament in the world. I'm with my people here," Sonego said. "I am so happy for this."
The 33rd-ranked Italian will have loud support from the stands when he returns to the court for his semi-final later on Saturday against top seed Novak Djokovic.
The tournament is organised by the 33-year-old's family with his younger brother Djordje acting as director, and will be played in his "Novak" facility where he regularly trains.
"Unfortunately, there will be no fans this year as we're in a difficult situation with a lot of people contracting the virus on a daily basis," the ATP 250 event's organiser Djordje Djokovic told Serbia's Nova S television.
"It's a very difficult year for everyone and the restrictions have resulted in a lot injuries. Regrettably, Borna Coric and Stan Wawrinka have had to pull out but it's still going to be a very strong field.
Bautista will play for a place in the final against 19-year-old Italian Jannick Sinner, the world No. 31 who beat Kazakhstan's Alexander Bublik 7-6 (7/5), 6-4.
In other men's action, John Isner, the 2018 champion and 2019 runner-up, booked a fourth-round spot with a 7-6 (7/5), 7-6 (7/5) victory over 11th-seeded Canadian Felix Auger-Aliassime.
Rublev, up 6-1, 5-2, was unable to convert three match points before the players had to leave the court for the second of two rain delays, but he wasted no time in closing out the match when play resumed.
The pandemic has already impacted the 2021 calendar, delaying the Australian Open and forcing the Indian Wells tournament in California -- the traditional lead-in to the Miami Open -- out of its usual slot in March.