The 26-year-old Monegasque driver, who has never managed a podium finish at his home race, clocked a best lap of one minute and 10.270 seconds to outpace Australian Oscar Piastri of McLaren by 0.154 seconds.
It brought Leclerc his third Monaco pole position and ended Verstappen's run of seven straight poles this year and a record-equalling eight in succession, shared with three-time champion Ayrton Senna.
Three-time world champion Verstappen struggled for Red Bull after hitting the barriers on his final fast lap.
"It was so nice," said a relieved Leclerc after a tense and thrilling qualifying.
"I'm really happy about my lap. The excitement was so high, but I know from the past that qualifying is not everything. The win is the target."
Leclerc's Ferrari team-mate Carlos Sainz was third ahead of Lando Norris in the second McLaren, Mercedes' George Russell and Verstappen.
"It was close," said Piastri. "If you took the second half of my first lap and the first half of my second one it would have been enough for pole!"
Sainz added: "I've struggled all weekend with confidence in the car so it's an improvement. I just wish I was fighting for pole."
Seven-time champion Lewis Hamilton was seventh in the second Mercedes ahead of RB's Yuki Tsunoda, Alex Albon of Williams and Pierre Gasly of Alpine.
In near-perfect Riviera conditions with sunshine and temperatures of 48 degrees celsius on the track and 21 in the air, Hamilton led Russell and the rest out of the pits.
Fiercely-contested
As the Mercedes ran two preparatory laps, it was Tsunoda on top for RB before Hamilton took command only for the rapidly-tumbling times to deliver Verstappen, Kevin Magnussen and Nico Hulkenberg on top, the two Haas drivers retaining their positions until Russell took over.
Leclerc, disappointed by Ferrari's decision to delay his entry, was 13th before he recorded a fast lap to go top in 1:11.653, a tenth clear of Magnussen and Hulkenberg.
In a fiercely-contested session, Hamilton jumped to second, but Perez was 14th and Norris 18th with only a minute remaining before, in a frantic finish to Q1, the Briton climbed to safety, but Perez was out along with Alonso.
The Mexican was 18th making an early exit for the second year running along with the two-time champion, in his Aston Martin, Williams' Logan Sargeant, Valtteri Bottas and his Sauber team-mate Zhou Guanyu.
"We've struggled all the way through," said Perez. "It is so tight, the margins I thought I could make the top five."
Norris was first out for Q2, but Sainz swiftly went quickest ahead of Verstappen in another tight tussle with barely a second separating the top 14 and the Alpines suddenly flying, Gasly claiming a place in Q3 in fifth. "Let's go, let's go!" he cried.
Piastri, looking confident, then clocked a 1:10.756 only to be beaten by team-mate Norris as the McLarens took control ahead of Leclerc and Russell before another busy finale saw Esteban Ocon of Alpine, Hulkenberg, Daniel Ricciardo of RB, Lance Stroll in the second Aston Martin and Magnussen eliminated.
Sainz was first out for the top ten shootout, followed by Hamilton who clocked 1:10.975, an impressive lap that was blown away immediately by Leclerc in 1:10.418, a lap nine-tenths faster than Verstappen's pole time last year.
The Dutchman's first timed lap left him third after losing time in the final sector. "Still the same issues in Turns Five and Ten," he reported.
As conditions continued to improve, Sainz and Verstappen chose to run again on a traffic-free lap, the rest following.
Hamilton improved to take fourth, but Russell beat him and then Verstappen glanced a barrier before Leclerc improved to 1:10.418 to take a dramatic pole ahead of Piastri, the Italian team's 250th pole position in Formula One.
