Bagnaia wins Japan MotoGP sprint to cut Martin's lead
Spaniard Acosta crashed with four laps to go while leading the field at Motegi, allowing defending world champion Bagnaia to take first ahead of Enea Bastianini and Marc Marquez.
Spain's Martin finished fourth and saw his overall lead over Italian Bagnaia in the championship standings cut to 15 points.
"I am very happy because with these conditions, it's not very easy to win and gain points," Bagnaia said after a sprint race that took place under heavy clouds with intermittent spots of rain.
"It's the maximum so thanks to my team again and let's keep working like this."
Four race weekends remain after Sunday's Grand Prix in Japan.
Pramac rider Martin is looking to win his first MotoGP championship after finishing second behind Ducati's Bagnaia last year.
Acosta was looking for his maiden sprint win after taking pole for the first time in the morning qualifying.
The 20-year-old lost the lead to Bagnaia at the first corner but regained it soon after and was pushing hard for the victory.
Instead, he slipped off his bike and let Bagnaia through, although the Italian still had to fight hard to hold off Bastianini and Marquez in a frantic finish.
Bagnaia finished just 0.181sec ahead of Ducati team-mate Bastianini and 0.349 in front of Marquez.
"The pace was very good for the sprint race," said Bastianini, who is third in the overall championship standings and shaved Martin's lead over him to 72 points.
"At the end, I was really close to Pecco (Bagnaia) but I lost time in the last two or three laps."
Martin started from 11th place on the grid but a tremendous start saw him move up to fifth on the first lap.
Marquez finished third after suffering a disappointment in qualifying when he set a new course record, only to see it wiped off for exceeding track limits.
Six-time world champion Marquez said he was able to put it behind him and "focus on the sprint race".
"I knew that I had the pace," said Marquez.
"In the last lap, I was pushing and catching them. It was a good fight with Enea."
Spain's Maverick Vinales, who started third on the grid, finished ninth after a disastrous first lap.
South Africa's Brad Binder, who went fastest in Friday's practice, did not finish.