"Oh my God I hit the wall two times," gasped Verstappen over the radio after crossing the line in the dying seconds with a best time of one minute 11.365 seconds to beat Alonso by a mere 0.084 seconds.
While Verstappen celebrated a lap for the ages, his teammate and closest title rival Sergio Perez crashed and will start last.
Ferrari's Charles Leclerc, who had been chasing his third home pole in a row, was third fastest but dropped to sixth after stewards found he impeded McLaren's Lando Norris.
Alonso made it a front row of double Formula One world champions and declared 'job done' at a street circuit where overtaking is extremely difficult but the second-placed starter has often prevailed.
The oldest driver on the grid, whose last win was a decade ago and who had not been on pole since he was at Ferrari in 2012, had taken the provisional pole with his own blistering last lap.
His first in the top 10 shootout was already impressive, the Spaniard telling his team he "drove like an animal" to achieve it, and the second was at times "over the limit".
But Verstappen delivered when it mattered with a barnstorming final sector in the afternoon sunshine, millimetre perfect and thrilling the capacity crowd with a final sector surge after lagging Alonso through the first two.
"We needed to pull that out the bag, but very lovely," he said as team boss Christian Horner hailed a "mighty" effort.
"I think that is one of the best laps he’s ever driven in qualifying," said Horner of his champion's 23rd career pole.
Alonso said the race was far from a foregone conclusion, with reliability a key consideration.
"I don't think that overtaking opportunities will come at all, as it's Monaco, but it's a very demanding race on the car, on the gearbox, on the brakes," he said.
French driver Esteban Ocon will start third for Renault-owned Alpine, and was briefly on provisional pole, with Spaniard Carlos Sainz fourth for Ferrari and seven times world champion Lewis Hamilton fifth in a revamped Mercedes.
The Briton had crashed in final practice, his Mercedes winched high overhead by a crane to remove it, and he was happy with where he ended up after twice coming close to an early exit from qualifying.
"The last time I enjoyed the track this much was 2007, 2008 and my Formula Three days here, I’ve generally really enjoyed it," he said.
NIGHTMARE FOR PEREZ
Alpine's Pierre Gasly lines up seventh, with Mercedes' George Russell eighth, AlphaTauri's Yuki Tsunoda ninth and McLaren's Lando Norris 10th.
Last year's winner Perez, 14 points behind Verstappen after five races, crashed at the Sainte Devote first corner in the opening session.
The Mexican also crashed in the final phase of qualifying last year but started third, and ahead of Verstappen.
This time the damaged Red Bull car was craned away, with Perez's hopes in tatters and the possibility of needing a new chassis.
Monaco was a race he had high hopes of winning again, with most of his previous victories secured on street circuits, but just scoring points is the target now.
"It’s going to be a nightmare tomorrow," he told reporters.
Both the Haas cars of Kevin Magnussen (17th) and Nico Hulkenberg (18th) failed to make it through the first session in what will be the team's 150th race.
Aston Martin's Lance Stroll was only 14th on the grid and behind the Williams of Alex Albon in 13th.
