An error-strewn game that never built any real momentum included seven missed penalties - four of them by Stormers fullback Joe Pietersen - before flyhalf Peter Grant came off the bench to convert the decisive kick at the end.
The Sharks dominated the first half, forcing the Stormers to make 62 tackles to the Durban side's 24 as they showed more invention with the ball in hand.
But a galvanised Stormers unit converted three penalties to one in the second period to clinch a tight game.
Grant, in his first game since returning from a stint in Japan, slotted the crucial penalty with two minutes remaining.