Reigning champion Bagnaia wins thrilling Spanish Grand Prix
Pole-sitter Marc Marquez, a six-times MotoGP champion, finished second for Gresini Racing at the Circuito de Jerez, with VR46 Racing's Marco Bezzecchi third.
Martin retained the lead in the driver's standings.
The Spaniard, on 92 points after four rounds, has a 17-point lead over second-place Bagnaia, whose Ducati teammate Enea Bastianini is a further five points behind in third.
Bagnaia, who started on the third row in seventh place, put himself in the mix early on with a series of audacious overtakes on the first lap, zipping past Bezzecchi, Martin and Marquez to take the lead.
Saturday's sprint winner Martin quickly wrested back the lead and was in front for much of the first half of the race, but the Spaniard lost the front end of his Ducati on the 11th lap and was dumped in the gravel.
"The most important thing was the first lap, after that I just tried to push and Jorge committed a mistake by going too strong on the brakes," Bagnaia said.
"I enjoyed a lot, every time I fight with Marc, I know I have to be aggressive. It was very good."
The second half of the race provided a different challenge for Bagnaia as he went from doggedly chasing the lead to defending it with Marquez pushing him hard in the last few laps.
Marquez, roared on by the fans in Jerez, twice overtook Bagnaia but Ducati's twice world champion made a late surge to claim his first win of the season.
"It was a tight fight, like old times you know," said Marquez, whose shoulder was covered in tyre marks following a particularly close encounter with Bagnaia.
"This is good for racing, he was super good and did the fastest lap of the race I think. I tried to follow him afterwards, but he had a bit more than me.
"I'm super happy for this first podium."
It was a tough day for the veterans on the grid, with Red Bull KTM's Dani Pedrosa crashing out early, before Aprilia's Aleix Espargaro and Johann Zarco of Honda LCR made contact and skidded off the track.
Pramac's Franco Morbidelli and Red Bull KTM's Jack Miller also crashed out after a coming together.
