Martin appeared to have the race wrapped up but made a mistake and fell at turn one on the penultimate lap to hand the win to the double world champion.
Marc Marquez, who will be Bagnaia's teammate next season on the factory Ducati, took second, with Alex Marquez, his younger brother, completing the podium.
"I don't know what happened," lamented Martin.
"It took me by surprise, I was pretty much in control, the pace was good, I was riding perfectly."
Martin's dramatic departure from the Sachsenring stage gifted not only the race victory to Bagnaia but also the riders' standings.
Bagnaia heads into MotoGP's summer break 10 points clear of Martin.
"It wasn't easy," said Bagnaia, who was winning his fourth successive grand prix after Assen, Italy and Catalunya.
"I saw Jorge was starting to make some mistakes on the last two or three laps, then when he had his crash I just slowed down a lot. (I'm) very happy. It's the first time I win here in Sachensring."
Marc Marquez is an eight-time winner on this circuit but the six-time former world champion has not won a race since October 2021.
"I promise you I would trade a victory to be on the podium with my brother, it's an amazing feeling" said the Spaniard.
His younger brother and current Ducati Gresini teammate Alex reflected: "It's a super amazing Sunday here, I gave everything.
"I did my own race and in the end with Martin's crash we were on the podium and I said, okay, we need to stay, we need to bring this home."
Costly mistake
The day had been going so perfectly for Martin up to his costly mistake - the winner of Saturday's sprint race holding a 15-point cushion at the top of the standings going into this ninth round of the season.
The pole-sitter held off Miguel Oliveira at the start, with Bagnaia moving up into second by the end of lap one.
'Pecco' then hunted down and went past his prey, Martin, on the following lap, but was unable to pull away and Martin was back in control of the race with 23 laps to go.
The Spaniard, on Ducati's satellite Pramac bike, did what Bagnaia was unable to do - put daylight between the pair.
With a third of the 30-lap contest remaining Martin was over half a second clear of his Pramac teammate Franco Morbidelli, with Bagnaia relegated to third.
But only briefly, as the double world champion nipped past Morbidelli with Martin now almost a second away up the road.
Bagnaia bided his time, and his patience paid off as Martin slithered out of the race and off the top of the standings with an error under pressure.
So it is Bagnaia and not Martin who heads off to MotoGP's summer holidays bathed in sunshine with the next race in their compelling title battle coming in Britain at the start of August.
