Despite little quality in the opening half-hour, the home side took the lead with their first shot on target.
When a Jack Hendry header bounced back off the post, both Oli McBurnie and Ryan Christie were on hand to bundle it home, with the latter getting the final touch.
Slovenia quickly went on the attack for an equaliser, but staunch Scottish defending denied them.
In the 38th minute, Sturm’s excellent reverse ball split the home defence, but Zan Vipotnik’s slight hesitance to shoot allowed Andy Robertson to make a heroic block.
Within 12 seconds of the restart, Vipotnik was again denied by last-ditch defending, as Hendry flung himself in front of the Swansea striker’s shot.
Time and again the away side found space behind the Scotland defence, and in the 57th minute they eventually converted a promising position into an equaliser.
Aljosa Matko’s run beyond the last defender was found by a pinpoint Sturm pass, and the striker’s low drive from the edge of the box was too accurate for Angus Gunn in the home goal.
Sebastien Pocognoli was unable to plug the gaping holes in his side’s defence, and 14 minutes after the equaliser the comeback was complete.
Roles were reversed, as Matko raced in behind the defence and squared the ball to Sturm, who thumped it home to seal a miserable descent into hopelessness for the hosts.
A huge win for Bostjan Cesar’s side means one win from their two November matches will be enough to seal a place in the promotion play-offs, while Scotland will now need to win both of their remaining games, including a trip to table-toppers Switzerland, to have any chance of joining Europe’s elite.