Reds standout Samu Kerevi took 90 seconds to exploit the overlap with a lovely cut-out pass before he backed up to score the game's first.
The Reds were through the line again thanks to speedy halfback Tate McDermott minutes later, with hooker Brandon Paenga-Amosa claiming the five-pointer for an eventual 14-0 lead.
Kolisi's return coincided with the Stormers' first try, Kobus Van Dyk powering over from close-range next to the upright as the game opened up.
McDermott duly sniffed out his own five-pointer from a quick tap and - notwithstanding a Damien De Allende try for the Stormers - Bryce Hegarty booted the Queenslanders out of reach with a penalty goal.
Coach Brad Thorn was proud of his side but was again left asking them to find the middle ground between expansive and safe rugby.
"Lots of positive rugby out there, lots of endeavour and there's a part of me that loves that," Thorn said.
"There's a composure around that ... often our biggest challenge is just us."
The Reds now move to 3-4 ahead of a two-game trip to the Republic, although they will be without winger Filipo Daugunu (broken arm), a host of Australian under-20s representatives and most likely a rested Kerevi.
The Reds' task was made easier against the Stormers by the absence of the injured Eben Etzebeth and fellow Springboks star Pieter-Steph du Toit, who flew home for the birth of his child.
It didn't slow the visitors early though, who were unlucky not to draw first blood through Damian Willemse and Sergeal Petersen.
Stormers coach Robbie Fleck rued his side's wastefulness.
"It's pretty tough to swallow that and those first 10 minutes (with Kolisi sin-binned), the Reds were pretty good at executing that advantage," Fleck said.
"We're not really proud of that effort in the second half - too many errors, too many missed tackles."