Hoping to avenge last season's knockout loss, the South African side started with promise, dominating early possession and taking a 3-0 lead via a Sacha Feinberg-Mngomezulu penalty.
But their momentum faltered after Seabelo Senatla was shown a yellow card for a cynical infringement at the breakdown. Glasgow quickly took advantage, with tries from Rory Darge and Kyle Rowe flipping the game on its head.
Despite showing strength in the scrums and set-pieces, the Stormers struggled elsewhere. Uncharacteristic errors from the backline, poor handling, and lazy defending after scoring undermined their efforts throughout the match.
They stayed within touching distance at half-time thanks to a second Feinberg-Mngomezulu penalty and a Senatla try, trailing just 19-13 after Henco Venter had added Glasgow's third.
Senatla's second try shortly after the break gave the Stormers renewed hope, but once again Glasgow responded instantly with Rowe's second score.
The Warriors tightened their grip on the match through a George Horne try and penalty, stretching the lead to 36-18 and leaving the Stormers with no route back.
The Stormers' night worsened when captain Salmaan Moerat left the field for a head injury assessment, further disrupting their defensive structure.
With the victory, Glasgow advance to the semi-finals, where they will face either Leinster or Scarlets.
For the Stormers, it's a painful case of deja vu - another season ended at Scotstoun, and another opportunity missed.