The Stormers have lost their past 10 games in New Zealand, and have just one win from four matches this season.
The late loss of hooker Ash Dixon, ruled out after a head knock at training, had little effect on the Highlanders, with Liam Coltman slotting in efficiently.
Coming off the bye, their scrum was solid and their work at the breakdown was relentless. While handling errors meant second-half tries went begging, the Highlanders always looked in control.