The Stormers held off a late fightback from the Reds to record a hard-fought 25-19 victory at Newlands.
Fly-half Damian Willemse kicked the home side ahead with an early penalty and was also involved in the move that led to Dewaldt Duvenage scoring the game's opening try.
Wilco Louw then fought his way over from close range to extend Stormers' lead, with the successful conversion making it 15-0 just before the half-hour mark.
Willemse added a further penalty but tries from Caleb Timu and Alex Mafi either side of the half-time break put the contest back in the balance in Cape Town.
Crucially, though, Stormers were the next to touch down, Raymond Rhule collecting Craig Barry's grubber kick to stretch his team's lead to nine points.
While Reds winger Filipo Daugunu scored a stunning solo try in the 80th minute to set up a frantic finish, Stormers stood firm to move clear of the Sharks in second place in the South African conference.