Mexican Perez needed to finish in the top three places to keep the title officially open for another day, with the main grand prix on Sunday, but started eighth and dropped to 11th before tangling with Alpine's Esteban Ocon on lap 11.
The Red Bull was then stuck in the gravel, with Perez stepping out.
That meant Verstappen, who was running in third place at the time and went on to finish second and 1.8 seconds behind McLaren's Oscar Piastri, was the champion for the third year in a row and while still racing.
"Max, you are a three-time world champion," team boss Christian Horner said over the radio as he crossed the finish line.
"That's unbelievable. It's been an incredible year for you."
Verstappen, who has won 13 of 16 grands prix so far including 10 in a row, replied: "Yeah, unbelievable guys. I don't know what to say. Thank you for providing me with such a car. It's been a lot of fun this year."
He also became the second youngest triple world champion after now-retired German Sebastian Vettel, who secured the third of his four crowns with Red Bull in 2012 at the age of 25, and the first to win the title in a sprint event.
The last to take the title on a Saturday was Brazilian triple champion Nelson Piquet in the 1980s.
Piastri's success ended Red Bull's domination of the sprints this season and earned the Australian rookie his first victory in F1, even if not a proper grand prix for the record books.
"When he (Verstappen) got through to second I thought I was going to be in a bit of trouble, but the pace was good. First Sprint win sounds pretty cool," said Piastri, who finished 1.871 seconds ahead.
NORRIS THIRD
His teammate Lando Norris finished third with Mercedes' George Russell fourth and teammate Lewis Hamilton fifth after starting 12th and cashing in on the collisions ahead of him.
Ferrari's Carlos Sainz was sixth with teammate Charles Leclerc seventh at the flag but then handed a five-second penalty for exceeding the track limits and demoted out of the points.
That moved Williams' Alex Albon up to seventh with Fernando Alonso eighth for Aston Martin.
While Piastri led away into the first corner on the medium tyres, with Russell using his softs to slot into second, Verstappen on mediums dropped initially to fifth and Perez to 11th.
That all but ended the contest, with Perez making no progress through the field.
Those on the mediums had the last laugh, however, as the softs wore down and Piastri passed Russell to take back the lead on lap 11.
The safety car was kept busy throughout, deployed shortly after the start when AlphaTauri's Liam Lawson went off at turn two and his car became stuck in the gravel.
It came in at the end of lap two, enough time for Russell to take the lead before Logan Sargeant also spun his Williams into the gravel a lap later and again triggered a safety car deployment.
The safety car came out a third time to deal with Perez's accident, leaving five laps of racing at the end and not enough time for Verstappen to catch Piastri.
The Dutch driver will start Sunday's main grand prix at Lusail on pole position.
