His puncture handed the win to his Mercedes team-mate Lewis Hamilton, who now heads the driver's championship by four points from Vettel in second.
Bottas took up the chase and received the lucky break he needed when the Red Bull crash caused a safety vehicle period, allowing him to pit more quickly and replace Vettel as leader.
Renault's Carlos Sainz was fifth and an incredibly unlikely sixth place for Alfa Romeo Sauber-Ferrari's Charles Leclerc who kept his nose clean amid the madness.
"These tyres are done", complained Hamilton, who pitted to switch to "softs" after 22 laps, rejoining third behind Bottas, but ahead of Verstappen.
During the race all went to the fact that Sebastian Vettel started on the pole and safely execute third victory of the season, but the scenario of the race changed dramatically after the collision the two pilots "Red bull". "If we can manage that, I think we've got a real good handling race auto". On the following lap, his right-rear tyre ripped apart after he drove over some debris sitting just in front of the start-finish straight, forcing him to retire from the race.
Lewis Hamilton is relieved Mercedes has managed to cut the deficit to Ferrari after struggling through much of practice in Baku. Hulkenberg's race came to an end when he lost the rear of his vehicle at Turn 5 and hit the wall, while Sainz dropped down the order with an early pit stop but eventually went on to finish fifth following retirements further ahead.
Opportunistic Force India: Sergio Perez just has a habit of picking up podiums in Formula One. Hamilton and Bottas followed him into Turn 1.
Verstappen's auto was lifted briefly into the air and both battered Red Bulls skated down the escape road into retirement. With just five laps to go, Mercedes chose to keep Bottas on track with his original set of tires, hoping the excessive caution laps were enough to extend the tire life.
After the final safety vehicle spell, the Ferrari made one last effort to reclaim the lead - and locked up, veering wide and allowing Lewis Hamilton into P2.
Romain Grosjean crashed under the safety vehicle on lap 42, sliding into the wall while weaving in an attempt to heat up his tires.
Hamilton's lack of a pace advantage on fresh, but harder, rubber allowed Vettel to stay out until lap 30 before taking on softs, but Bottas ran even longer. It was his first win since last October.
Canadian Lance Stroll, third in Baku past year, will start 10th for Williams - the only team yet to score a point this season.