Red Bull's championship leader was down in fifth after the first barrage of Q3 runs. Amid scintillating pole bids by Esteban Ocon, Charles Leclerc and Alonso, Verstappen then had the last word as the chequered flag dropped.
Despite two personal bests, he was still trailing Aston Martin driver Alonso heading into the final sector, but with a stunning recovery, his 1m11.365s effort pipped the Spaniard at the line by 0.084 seconds.
Verstappen was seen brushing the pit straight wall on his way to P1 and later revealed he "clipped a few barriers" on his way to his first-ever pole in the principality.
"My first sector was not ideal on my final lap, I think Turn 1 was a bit cautious," he said.
"But then I knew I was behind, so the last sector I gave it everything I had, clipped a few barriers but of course very happy to be on pole here for the first time.
"We knew that this was going be a bit of a struggle for us to get everything together, yesterday was not the best start but we kept on improving and kept on being better. But then in qualifying you need to go all out and risk it all."
Sunday's race provides a prime opportunity for the Dutchman to extend his lead on Red Bull team-mate and nearest rival Sergio Perez, who crashed in Q1 and will start at the back of the grid.
Verstappen vowed to "keep it clean and calm" and steer clear of any chaos the narrow street circuit might throw at him.
"We need a clean start, it's a short run to Turn 1. In Monaco a lot of things can happen, right? Safety car, rain, you name it. There's always a bit of chaos involved.
"I think race pace wise the car is quick, that is not the problem, but we just need to keep it clean and calm."
Team boss Christian Horner says Red Bull will have to "check every corner" of Verstappen's car after his wild romp to pole.
"Absolutely. I think we're going to have to check every corner, every wheel rim that he's used today," he told Sky.
"He was pretty excited when he crossed the line of how hard he touched the barrier. So, we were lucky that we weren't sweeping it up after that last corner.
"Hell of a lap from him and well deserved."
