The pole was Leclerc's third in Monaco and 24th of his career, but he has yet to stand on the podium in front of his home crowd.
Red Bull's triple world champion Verstappen, who was chasing a ninth successive pole and eighth of the season, hit the wall and will line up only sixth. McLaren's Oscar Piastri joined Leclerc on the front row with Ferrari's Carlos Sainz and McLaren's Lando Norris together on the second row.
"We need to put everything together for Sunday's race," said Leclerc, who can become the first Monegasque to win since Louis Chiron drove a Bugatti to victory in the early years of an event that now stands for glamour and history.
"In the past here we didn't manage to do so, but we are in a stronger position and we are a stronger team. I'm sure we can achieve great things tomorrow and the win is the target."
Red Bull's triple world champion Verstappen, who was chasing an unprecedented ninth successive pole and eighth of the season, will line up only sixth after pushing too hard and paying the price.
"I've hit the wall," the championship leader, who has won five of seven races so far this year, exclaimed over the team radio.
Verstappen had been third fastest after the initial round of fast laps in the final top 10 shootout, with Leclerc setting a time of 01:10.418 seconds and Piastri 1:10.444.
Leclerc improved on his time with his final effort of 1:10.270, while Piastri also shaved 0.020 of a second off his previous best.
Verstappen, who has been wrestling with his car since Friday practice, threw everything at it but his mistake effectively cost him three places on a circuit where overtaking is very difficult and races are often processional.
Mercedes George Russell will share the third row with Verstappen. Seven times world champion Lewis Hamilton secured seventh place, a disappointment after the Briton's strong showing in practice, with RB's Yuki Tsunoda eighth.
Alex Albon qualified ninth for Williams and Pierre Gasly a morale-boosting 10th for struggling Renault-owned Alpine in the closest thing to a home race for the Frenchman.
Verstappen's teammate Sergio Perez was the big casualty of the first phase of qualifying with the Mexican set to line up only 18th.
Aston Martin's Fernando Alonso, another past Monaco winner, also had a bad day and starts 16th.
