His time of 01:41.697 completed a pole double in Baku for the Monegasque, who also starts from the top slot in Sunday's grand prix after setting the fastest time on Friday.
Mexican Sergio Perez joined Leclerc on the front row for the 17-lap (100km) sprint, 0.147 off the pace, with Red Bull teammate and championship leader Max Verstappen lining up third and George Russell fourth for Mercedes.
"Ferrari needs to be on top and I will do everything to win," said Leclerc, who has scored only six points from the first three races.
The Ferrari driver hit the barriers nose-first at turn five, spoiling the final lap for teammate Carlos Sainz, but got the car back to the pits to park up behind the polesitter's marker board.
"The rear (tyres) overheated on the second lap, I tried to push a bit more to gain some lap time and lost it," said Leclerc.
Verstappen, winner in Baku last year and also of two of this season's races, said he had a "big moment" in the final phase at turns five and six that overheated the tyres.
"A bit unfortunate to be P3 but with this heat and how the tyres are behaving I think it's not going to be a straightforward sprint," he added.
"Some tracks you can just set off and it's fine, here maybe a bit of tyre management comes into play so there's still all to fight for."
Sainz qualified fifth, with seven times world champion Lewis Hamilton in sixth for Mercedes and Alex Albon seventh for Williams.
The Aston Martins of Fernando Alonso and Lance Stroll were eighth and ninth, with the Spaniard suffering recurring issues with his car's drag reduction system (DRS).
McLaren's Lando Norris will line up 10th after being unable to take part in the final eight minutes because he had no new soft tyres remaining after using his allocation on Friday in a strategic decision by the team.
The move appeared to have backfired when teammate Oscar Piastri, who did have a set of soft tyres, was kept out of the final shootout by his teammate and qualified 11th.
The shootout, described by Perez as "fast and furious", is a new format aimed at adding to the excitement by replacing what had been a low-key final practice.
Formula One hopes it will encourage drivers to race hard in the sprint without having to worry about grid positions for the next day. Points are still awarded to the top eight.
The first 12-minute phase of the session was red-flagged with 25 seconds remaining when rookie Logan Sargeant crashed his Williams at turn 15.
"Man, the Ferraris were in the middle of the road," complained the American, whose mechanics have another race on their hands to get the car ready in time for the sprint race.
The red flags meant nobody else could improve their time, with AlphaTauri's Japanese driver Yuki Tsunoda the big loser as they came just as he was about to cross the line and move out of the bottom five.
Pierre Gasly's nightmare weekend continued, after a fire in practice and a crash in Friday qualifying for Sunday's main race, when the Alpine driver pulled into the pits with a suspected exhaust leak. He qualified 19th.
