Lewis Hamilton will start his bid to become a seven-time world champion from sixth on the grid. Hamilton, so often the master of the slippery conditions, was an eye-watering 4.7 seconds slower than Stroll.
Hamilton's Mercedes team-mate Valtteri Bottas, the only driver still mathematically in the title race, finished ninth. If the order remains that way here on Sunday, Hamilton will match Michael Schumacher's championship haul.
"I cannot put it into words right now," said Stroll, 22, who missed the Eifel Grand Prix last month through illness and subsequently tested positive for coronavirus.
"I am shocked and I did not expect to be up here. We didn't look too competitive but I am so happy now.
"I put that lap together and I was under so much pressure. I had the confidence in the car and I just nailed every corner and it is such a nice way to bounce back after a few rough weeks. It is the highlight of my career. This is what we dream about."
Verstappen has impressed all weekend on F1's first visit to the Istanbul Park Circuit since 2011 and he looked set to claim his third career pole.
But Racing Point pulled off a masterstroke by putting both of their drivers on the quicker intermediate tyre for the start of Q3. Verstappen changed from the extreme wets to the inter tyres but could not match Stroll's pace.
Stroll's shock pole here ends Mercedes' run of starting from pole at every round of this Covid-disturbed campaign.
Hamilton needs to finish ahead of Bottas to be sure of the title, and although he will be staggered to be so far off the pace, the six-time world champion, who went over to congratulate Stroll, will no history still beckons for him on Sunday.
Qualifying took the best part of two hours to complete after Q1, the opening 18-minute running, was suspended on two occasions.
After just 10 minutes of action, race director Michael Masi deemed the conditions were too dangerous as rain continued to lash the venue 30 miles east of Istanbul.
A 45-minute delay ensued as Bernd Maylander pounded round the track in his safety car while a truck was also sent out in a bid to remove some of the standing water.
Masi gave the green light for qualifying to resume at just before 4pm local time, but the session was suspended only three-and-a-half minutes later when Romain Grosjean beached his Haas at the opening bend.
With Grosjean's Haas removed Q1 eventually finished, one hour and 10 minutes after it started.
Stroll then provided a major upset in Q3, becoming the first Canadian pole-sitter since Jacques Villeneuve 23 years ago.