McLaren had hoped to secure their 10th Formula 1 constructors' title, and second in a row, on Sunday with a record seven rounds remaining, but must now wait until Singapore on October 5th.
George Russell finished second for Mercedes, who moved up to second place and 333 points behind McLaren with 346 still to be won, and Carlos Sainz third for Williams's first podium finish of the season.
Norris started and finished seventh, with his chances of scoring more points scuppered by a slow pitstop for the second successive race.
The win was Verstappen's second in a row, and he was absolutely dominant, leading from pole to flag and setting the fastest lap for a "Grand Slam" that revived his slim title hopes with the reigning champion now 69 points behind Piastri.
"I think this weekend has been incredible for us," said Verstappen after his 67th career win and fourth of the season.
"For us to win here again is just fantastic.
"We had clean air all the time, and you could then look after your tyres. And yeah, it was pretty straightforward."
Russell, recovering from sickness, was happy to see a rough weekend finish strongly, while Sainz was the happiest man on the podium.
"I cannot describe how happy I am and how good this feels. It tastes even better than my first ever podium," said the four-time race winner with Ferrari.
The big story came on the opening lap when Piastri, who crashed in qualifying and started ninth, jumped the start and went to the back of the field before ploughing into the wall at turn five.
Piastri crashes out early
The uncharacteristic errors ended the Australian's record of being the only driver to score in every race this season and also put the brakes on a run of 34 races in the points.
He had not failed to finish a race since his 2023 rookie season.
"Certainly not my finest moment," the 24-year-old told Sky Sports.
"I just anticipated the start too much and yeah...silly simple error really."
Italian rookie Kimi Antonelli was fourth for Mercedes, redeeming himself after a poor home race weekend at Monza, with New Zealander Liam Lawson enjoying a career-high fifth for Racing Bulls.
Yuki Tsunoda was sixth for Red Bull, with Ferrari's Lewis Hamilton and Charles Leclerc following Norris home in eighth and ninth, with the pair swapping places on team orders because Hamilton had fresher tyres.
French rookie Isack Hadjar made it a double points finish for Racing Bulls in 10th.
Piastri's crash brought out the safety car, a regular feature of races in Baku, and provided the main drama of the 51-lap race around the windy city streets, where the threat of a downpour failed to materialise.
Norris lost a place to Hadjar and was then passed by Leclerc when the race restarted after the safety car period.
The Briton followed Leclerc past Hadjar, whose car had suffered a hydraulics problem before the start, and could have passed both Leclerc and Lawson had his pitstop not cost him critical seconds.
He ultimately passed Leclerc with 10 laps remaining but was unable to find a way past Tsunoda.
"I did the best I could yesterday, the best I could today," said Norris.
"Every race I didn't win was an opportunity missed, so of course, today I wanted more. It was not a good result, but I couldn't do anything more today."