Leclerc stormed to his sixth pole of the season with a sensational lap that was 0.282s faster than Perez, who was again the quickest Red Bull driver in Baku after jumping his teammate with a last-gasp improvement to take second.
Reigning world champion Verstappen, who leads the world championship by nine points over Leclerc heading into Sunday’s race, ended up 0.347s adrift of the Monegasque’s pole time.
Carlos Sainz had sat on provisional pole after the opening salvo of Q3 but a scruffy final lap from the Spaniard, coupled with improvements from his rivals, dropped the Ferrari driver to fourth and nearly half a second down.
Ferrari and Red Bull were in a league of their own around the streets of Baku, with George Russell 1.3s off the pace in fifth for Mercedes.
Pierre Gasly put his AlphaTauri sixth to split the Mercedes drivers, with Lewis Hamilton qualifying seventh.
The seven-time world champion could yet be hit with a penalty for driving unnecessarily slowly in qualifying, an incident which is under investigation.
The second AlphaTauri of Yuki Tsunooda was eighth, ahead of Sebastian Vettel and Fernando Alonso, who completed the rest of the top 10 for Aston Martin and Alpine.
Lando Norris missed out on Q3 by just 0.022s as he finished 11th and 0.011s ahead of McLaren teammate Daniel Ricciardo.
Esteban Ocon took 13th, while F1 rookie Guanyu Zhou outqualified Alfa Romeo teammate Valtteri Bottas for the first time this season.
Kevin Magnussen was 16th ahead of the Williams duo headed by Alex Albon, who was left furious at Alonso for ruining his final flying lap when the Spaniard ran down the Turn 15 escape road.
“He has to get penalised this guy,” Albon fumed over team radio. “He braked so early and just went off the circuit.”
Behind the second Williams of Nicholas Latifi, Aston Martin’s Lance Stroll was knocked out in 19th after crashing twice in two laps.
Having already hit the Turn 7 barriers, the Canadian ended up ended up in the wall at the exit of Turn 2 as he caused the first red flag of qualifying.
The struggling Mick Schumacher was slowest of all in 20th for Haas.