The top three in the qualifying shootout for the 19-lap race were separated by just 0.069 of a second, with Mercedes' Lewis Hamilton third fastest.
Verstappen led the way in the final session, with each driver having one run on soft tyres, and set a best time of one minute 34.538 seconds.
McLaren's Lando Norris slotted into second place but Leclerc, who will start Sunday's main race from pole position with Verstappen only sixth after having his best lap deleted in Friday's qualifying, pushed him off the front row with a time 0.055 seconds slower than Verstappen.
Hamilton then crossed the line and pushed Norris further back to fourth.
McLaren's Australian rookie Oscar Piastri, sprint winner in Qatar from pole two weeks ago, was fifth fastest, with Ferrari's Carlos Sainz sixth and Red Bull's Sergio Perez seventh.
"It's going to be an exciting afternoon," said Verstappen, who clinched his third title in Qatar.
"You can see a lot of different cars being very close to each other, so you don't know what's going to happen in the Sprint Race."
Only the top eight in the sprint score points.
The sprint pole was Verstappen's third in five such races this season.
