The Stormers had come to Scotstoun looking for revenge after being beaten at this stage in the same venue last season.
Feinberg-Mngomezulu kicked the first points of the match as the visitors came out the blocks fast, but Glasgow responded quickly.
Stormers wing Senatla was sent to the bin for cynically killing Warriors’ momentum after a lovely break from Kyle Steyn. Glasgow kicked to the corner and an innovative lineout move created space for Darge to dive over.
That energised the Scotstoun crowd and they soon had more to cheer when Sione Tuipulotu stabbed a clever grubber in behind for Rowe to latch on to and score.
Stormers then hit back in bizarre circumstances.
Feinberg-Mngomezulu launched an up-and-under from a free-kick that caused chaos in the Warriors 22. The ball bounced kindly for Senatla, just back on after his yellow card, to go in under the sticks.
Again, Glasgow’s response was swift, Venter peeling off the base of a ruck to power over in the corner.
Warriors had to withstand a big Stormers onslaught late in the first half to go in 19-13-up at the break, but they were undone early in the second half as some sharp handling in the backline sent Senatla streaking away for his second try to bring the visitors back within one.
This was turning into a URC classic and the breakneck pace showed no sign of slowing up.
Tom Jordan found a kink in the Stormers defence and burst through before feeding Rowe to go over for his second try of the evening.
Now Glasgow had to buy themselves some breathing space on the scoreboard and they did it in the most glorious fashion.
Euan Ferrie snatched a Stormers lineout and off Glasgow went from deep. A succession of brilliantly executed passes ended with Rowe feeding Horne to scamper away for Warriors’ fifth try.
That ended the South Africans’ resistance and Glasgow saw it out in relative comfort to ensure their bid for back-to-back URC crowns remains on course.
Glasgow Warriors head coach Franco Smith told Premier Sports: “We were a little bit off it [at the end of the regular season] but I’m glad we could bounce back.
“It’s good to have played Friday night. We get to wait and see and whatever comes our way, we will make sure we recover well.
“Good people make good players. We tried too hard to finish second in the competition and the weight of playing the semi-finals at home became too much in the end.
“It’s a different competition. It’s knockout rugby. It brings more excitement. The boys have learned and adapted.”
Glasgow Warriors: McKay, Rowe, McDowall, Tuipulotu, Steyn, Jordan, G Horne, Bhatti, Hiddleston, Walker, Samuel, Cummings, Ferrie, Darge, Venter.
Replacements: Matthews, McBeth, Talakai, Williamson, Mann, Duncan, Dobie, Hastings.
Stormers: Gelant, Loader, Hartzenberg, D du Plessis, Senatla, Feinberg-Mngomezulu, Ungerer, Vermaak, Venter, Fouche, Moerat, van Heerden, de Villiers, Theunissen, Roos.
Replacements: Dweba, Matongo, Sandi, Schickerling, Evans, L Nel, Jantjies, Libbok.
Referee: Andrew Brace (IRFU).
