Bagnaia's teammate Enea Bastianini finished second, while six-time MotoGP champion Marc Marquez of Gresini completed the podium in third.
Pramac rider Martin slid off on the first lap to gift reigning champion Bagnaia the race lead in a blow to his championship hopes with Sunday's grand prix and five race weekends to go.
Martin had secured pole position at the sweltering Mandalika track on Lombok island with an all-time lap record in qualifying but could only claw his way back to 10th place after the costly incident.
A win in the sprint would have guaranteed Martin the championship lead heading into the next round in Japan next weekend.
Marquez, who has pulled himself back into the title picture in recent weeks, made an incredible early surge from 12th to finish third.
Marco Bezzecchi, who survived a crash in qualifying to secure a front-row place on the grid, finished fourth.
MotoGP returned to Indonesia in 2022 for the first time since 1997, at a new venue on the resort island of Lombok.
