The match, played at a blistering pace, saw the home side edge ahead 26â-24 at half-time, with both teams trading tries in a high-scoring encounter.
Stormers utility back Sacha Feinberg-Mngomezulu was the standout performer, racking up 24 points with a hat-trick of tries, three conversions, and a late penalty that ultimately sealed the win.
Feinberg-Mngomezulu's performance was nothing short of electric. He executed a clever chip-and-chase to complete his hat-trick, while his audacious pass set up Suleiman Hartzenberg's try in the corner.
Evan Roos added to the tally with a powerful surge from the base of the scrum for a try.
The Stormers had to overcome a yellow card to fullback Warrick Gelant and an unfortunate injury to veteran flanker Deon Fourie, who limped off just two minutes after coming on.
Connacht were in the fight throughout, with Shane Jennings bagging two tries on the wing. Tries also came from Conor Oliver, Dylan Tierney-Martin, and replacement Josh Murphy.
However, fly-half Jack Carty missed two early kicks, and centre Cathal Forde's yellow card for repeated infringements proved costly. Santiago Cordero was denied a try due to obstruction in the buildup.