The Spaniard's surprise victory on a clear afternoon in Texas was a first for his LCR Honda team since Britain's Cal Crutchlow in Argentina in 2018.
Italian Luca Marini (25), MotoGP great Valentino Rossi's half-brother, was second for the VR46 Ducati team and France's 2021 world champion Fabio Quartararo (23) completed the podium on a Yamaha.
VR46's Marco Bezzecchi (24) finished sixth to stretch his championship lead to 11 points over fellow Italian Bagnaia after three rounds.
Aprilia's Maverick Vinales and Miguel Oliveira finished fourth and fifth.
"We got the victory (in Austin) with Moto3, Moto2, Suzuki and then Honda," Rins said of his impressive record of success at the Circuit of the Americas.
Bagnaia, who had seemed certain to retake the championship lead with his second double victory of the campaign, threw away 25 points when he made an unforced error on the eighth of 20 laps and crashed - for the second time in two weekends.
In Argentina two weeks ago the Italian crashed out of the Sunday race while in second place.
"Today, just crashing like this, was something too strange," said Bagnaia, who had led away from the start with Rins doing his best to apply some pressure, of the turn two incident.
"Losing 45 points in two weekends is not a good thing."
Rins, who won in Austin with Suzuki in 2019, managed the remaining laps without incident for his sixth career win in the top class and first since he joined LCR Honda at the start of the season.
Team boss Lucio Cecchinello had reminded the Spaniard to keep calm in the closing lap and said he felt like he was in a dream.
"It was difficult to stay always with my eye on the television," said the Italian of his anxious pitwall experience.
"I tried to look around, look at the board and say something to the mechanics."
Spaniards Jorge Martin (25) and Alex Marquez (26) collided on the first lap, the Pramac Ducati rider losing the front of his bike and taking out Gresini Ducati's Marquez.
That completed a nightmare weekend for the younger of the Marquez brothers, the Spaniard also crashing on Saturday after vomiting in his helmet.
Aprilia's Aleix Espargaro (33) was another faller at turn 12 on the opening lap.
KTM's Australian Jack Miller (28) soared to third from 10th on the opening lap but then crashed on lap seven with Quartararo taking the podium place. The Frenchman was second after Bagnaia's crash before Marini came past.
Repsol Honda's Spaniard Joan Mir (25), the 2020 world champion for Suzuki, was another faller at the half distance. Teammate Stefan Bradl (33) and LCR's Takaaki Nakagami (31) also fell, leaving only 13 finishers.
