Crawford, who won alpine combined bronze at last year's Beijing Olympics, clocked 1min 07.22sec down L'Eclipse piste in the French resort.
"It's been a long time since I've dreamed of being on the top step at a world stage. I've been teasing myself with it the last couple of years with a couple of World Cup podiums and Olympic Games bronze," said Crawford.
"To be able to come out today and put down a run worthy of winning was an amazing experience. I'm going to cherish this day for a long time."
That time was enough to edge fancied Norwegian Aleksander Aamodt Kilde into silver by just one-hundredth of a second.
France's Alexis Pinturault, fresh from winning alpine combined gold on Tuesday, again thrilled his home Courchevel crowd by taking bronze, at 0.26sec.
With just three World Cup podiums to his name, albeit two in December of this season, Crawford was a rank outsider coming into the race.
His gold was Canada's first since Erik Guay claimed the super-G title in the 2017 worlds in St Moritz.
Starting with bib number 10, Crawford could only look on as first in-form Swiss Marco Odermatt, the runaway World Cup overall leader, streaked into the lead.
Odermatt was hot favourite, having notched up 13 podium finishes in his last 16 participations in World Cup super-G events, including seven victories - four of which have come this season.
Pinturault was next down, the flag-waving crowd going wild in a cacophony of sound mixing cowbells, airhorns, a sole trumpeter and bass-heavy dance music.
French hopes of a second gold in as many days was shortlived as Kilde next descended the course in bib number nine.
Pinturault was left shaking his head and grimacing in the knowledge that the Norwegian, winner of two World Cup super-G races this season, had taken the lead.
No one had bet on Crawford, however, the 25-year-old Canadian superbly negotiating speeds of 115km/h (71mph) and jumps of up to 45 metres down a course that demanded both technical and gliding ability.
That run pushed Odermatt off the podium, into fourth (+0.37).
Defending world champion Vincent Kriechmayr finished 0.87sec off the pace, the Austrian struggling with the "turny" bottom third of a race run in cold, sunny conditions.
Kriechmayr's teammate Marco Schwarz had the best chance late on to get Austria onto the podium, but a mistake two gates from the finish saw him lose invaluable time and eventually finish sixth, just one-hundredth of combined bronze medallist Raphael Haaser (+0.58).
