Rins took the lead from Bagnaia in the final lap and crossed the finish line ahead of six-times MotoGP champion Marc Marquez who grabbed his 100th podium as premier class motorcycle racing returned to the seaside circuit for the first time since 2019.
"Last time here with Suzuki, it's special," said Rins, who will join Honda satellite outfit LCR Honda in 2023. "So to go out with a victory is amazing."
Yamaha's Quartararo went wide at turn four early in the race to drop to 22nd place before crashing out after another mistake to cede the championship lead to Bagnaia whose victory charge was ended in dramatic circumstances by the top two late on.
"When I saw that Fabio was out, I said to myself 'a win is OK, but no problem if they overtake me in the final laps'," said Bagnaia, who holds a 14-point lead over Quartararo heading into the penultimate round in Malaysia next week.
"I'm very happy, we are now leading the championship."
It was heartbreak for local favourite Jack Miller, as his slim title hopes were crushed after a fall following contact with Honda's Alex Marquez.
Marquez had crashed in the warm-up as rain played havoc ahead of the race but the Spaniard had no problems in better conditions and celebrated his first podium finish since last year's Emilia Romagna Grand Prix.
"I enjoyed the race a lot, it doesn't matter the position," said Marquez. "I gave it everything I had. Step by step we are getting better."
Earlier, Spain's Izan Guevara (GASGAS Aspar Team) secured the Moto3 crown with a superb victory, taking himself more than the requisite 50 points clear of his nearest rivals with two rounds remaining.
