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.
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.
He 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.
Martin started from 11th place on the grid but a tremendous start saw him move up to fifth on the first lap.