Fourth versus fifth in the table and with a host of close and entertaining meetings in the past, this was a tough one to call.
Warriors were unbeaten at home all season, while the Stormers had picked up steam with four straight victories to end the regular season.
Glasgow stole a line-out on 10 minutes and some brilliant handling in the back-line sliced the Stormers open, but the final pass from Cancelliere was fumbled forward by Horne with the try-line waiting.
It was a golden chance and one the home crowd hoped would not come back to haunt their side.
As the wind and rain swirled around Scotstoun, Libbok was off-target with his first penalty attempt before Horne showed him how it was done to get Glasgow up and running.
Libbok pulled another penalty wide as the World Cup-winning fly-half struggled to find his range and the game as a whole, while ferociously physical, was struggling to get into any sort of flow as the slippy ball made handling difficult for both sides.
Horne banged over another penalty and, while the home side had to withstand a period of pressure near their own line, they went in at the break 6-0 to the good.
Glasgow’s Huw Jones scorched through a gap early in the second half but ignored the two support runners on his outside and the move fizzled out.
Stormers had barely existed as an attacking force and their cause was not helped when captain Salmaan Moerat was sin-binned for a head-on-head collision with Nathan McBeth.
The 14 men were undeterred and some slick back-line play led to Sacha Feinberg-Mngomezulu slipping Loader in to dive over in the corner. Libbok failed to convert and Glasgow remained ahead by one.
That Stormers try sparked the Warriors into life and they hit back immediately.
Sione Tuipulotu powered through four or five defenders thanks to a thunderous run, with Cancelliere on hand to collect the offload and run in for the score.
This slow burner of a quarter-final was exploding into life in the final 20.
After a loose line-out throw on their own five metre line, substitute scrum-half De Wet caught the Warriors fringe defence out with a show-and-go to dive over.
Incredibly, Libbok saw what should have been a straightforward conversion almost bang in front of the posts come crashing off the upright and Glasgow led by three.
Once again, the Glasgow defence was superb. A brilliant break from Tom Jordan took Warriors deep into Stormers territory and, after a long passage of patient phase play battering at the line, substitute Venter forced himself over.
Thompson pounced on a charged-down kick in the final play to dot down and cap a strong display in the final 20 to seal Glasgow’s passage into the semi-finals.
There, they will have a chance to avenge last season’s quarter-final defeat by Munster.
