Verstappen's woes ensure Ferrari hold the advantage heading into the race tomorrow, after Carlos Sainz set the third fastest time alongside Leclerc's pole.
If the Scuderia maintain their position and dominate around the streets of Baku, they could challenge their rivals in both the drivers and constructors' championship.
Verstappen's main title contender, Lando Norris , made a surprising Q1 exit in a blow to his championship chances, with the McLarenstar running wide at Turn 16 and aborting his lap.
The Brit looked set to make it into Q2 but a yellow flag caused him to back off and was eliminated in a shock early exit.
"Everyone did their second laps and I didn't," he said to Sky F1.
"It was unlucky, that's all. I had to lift.
"It is what it is. There's a long race ahead, we have some good tyres in the bank, I'm still hopeful we can get a good result. I think the car is quick. Bit frustrating, but nothing I can do."
Williams demonstrated impressive pace, with both Alex Albon and Franco Colapinto making the top ten.
However, the team made a major mistake by leaving a bright yellow airbox in Albon's car, as the team were set to produce their best qualifying result of the year.
Colapinto finished ahead of his team-mate, in a result that impressed team boss James Vowles.