The sprint victory after an engrossing duel moved Bagnaia to within four points of championship leader Martin as the Italian looks to mark his 100th MotoGP race on Sunday with Ducati's 100th victory.
"I tried everything to be at my 100% riding and luckily I'm able to ride at my 100%. I missed the start again and to overtake Jorge was very, very tough," Bagnaia said.
"I used his mistake to lead and after that moment everything was much easier. The pace today was incredible... I'm very happy. Thanks to my team that did an amazing job again."
Bagnaia had taken pole in qualifying earlier in the day with a lap record but was pushed back to third when Martin had a dream launch off the line, with KTM's Brad Binder right behind him before the reigning champion took back second place.
Marc Marquez crashed in qualifying and started on the third row but the Spaniard, who won the San Marino Grand Prix at the same circuit two weeks ago, quickly moved up to fourth behind Ducati's Enea Bastianini.
Martin maintained his lead despite Bagnaia's sustained pressure but with six laps to go, the Spaniard blinked and went wide, leaving Bagnaia to streak away.
"I couldn't keep that concentration when I got the track limit warning, a small mistake and I lost the race," Martin said.
"For tomorrow maybe I will be a little stronger."
As Martin looked to reel in Bagnaia, the Pramac Racing rider also had to look over his shoulder with Bastianini ensuring he was sandwiched between two red factory Ducatis after the Italian left Marquez far behind.
But Bagnaia maintained his composure and took the chequered flag less than three-tenths of a second clear of Martin, with Bastianini finishing third for an all-Ducati podium while Marquez finished a distant fourth.
Rookie Pedro Acosta finished fifth on his Tech3 bike after overtaking Binder, who finished sixth.
Fabio Quartararo was seventh for Yamaha, who had only one rider available after Alex Rins withdrew from the weekend with flu. Rins did not take part in qualifying either and is resting to recover in time for the Indonesian Grand Prix next weekend.