Leclerc improved on the benchmark on his second run with a time of 1:40.203, and while both Red Bull drivers also made gains, Verstappen was 0.188s behind after his final effort.
It marked a hat-trick of pole positions around Baku for Leclerc.
Perez wound up third, ahead of Carlos Sainz, who was a non-factor in the pole fight as he wound up eight-tenths behind his Ferrari team-mate.
Lewis Hamilton salvaged fifth place for Mercedes after only narrowly scraping through to Q3, while team-mate George Russell was only 0.004s behind in Q2, but was eliminated in 11th.
Aston Martin had a relatively subdued day by its lofty 2023 standards; Fernando Alonso was sixth while a DRS issue limited Lance Stroll to ninth spot.
McLaren had both cars in Q3 as Lando Norris was seventh, as Oscar Piastri rounded out the top 10, while AlphaTauri’s Yuki Tsunoda was an excellent eighth to secure his best grid spot of the year.
Williams displayed promise in Q1 but a minor error through Turn 4, combined with traffic, thwarted Alex Albon as he wound up 13th, while Logan Sargeant made Q2 for the first time but finished at the foot of the standings in that session.
Alfa Romeo had Valtteri Bottas in 14th and Zhou Guanyu 16th, either side of Sargeant, as the team’s one-lap struggles continued.
Haas suffered a double Q1 exit, with Nico Hulkenberg 17th, while Kevin Magnussen was compromised by a technical issue en route to 18th.
Magnussen attempted to return to action, following an engine complaint early in Q1, but was called back into the pits due to the intermittent nature of the problem and could not set another lap.
Alpine’s Pierre Gasly suffered a premature end to a problematic day as he crashed in Q1.
Gasly was left with limited running in practice following a dramatic engine failure but Alpine managed to ready his A523 with new components despite the short turnaround time.
However, Gasly hit the wall as he exited Turn 3, scattering the track with debris, and forcing the Frenchman out of the session.
Nyck de Vries’ tricky start to his rookie season with AlphaTauri continued.
De Vries, who was a promising sixth in practice, crashed on his first push lap after carrying too much speed into Turn 3.
That caused the session to be halted and the suspension was longer than usual – almost 20 minutes – after the front of de Vries’ AT04 became wedged in the TecPro barriers.
Saturday’s Sprint Shootout is scheduled for 12:30 local time