Two George Horne penalties gave Warriors a 6-0 lead at the break.
Ben Loader and Paul de Wet tries either side of a Sebastian Cancelliere score for the hosts gave the South Africans hope, but they were hindered by a hapless kicking display from Manie Libbok.
Tries from substitutes Henco Venter and Ross Thompson, allied to 12 points from the boot of Horne, took Glasgow into the last four.
Warriors will now face defending champions Munster in Limerick next Saturday.
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.
