Damian de Allende and Dillyn Leyds also crossed for the hosts, while the boot of Damian Willemse contributed 14 points in a victory that leaves Stormers third in the South African Conference, and ninth overall.
The Rebels remain second in the Australian Conference thanks to their strong start to the season, and are seventh overall, but have now lost their last four matches.
After Rebels’ Jack Debreczeni sent over two penalties either side of one from Willemse early on, De Allende burst through a tackle and then advanced beyond the visitors’ defence to open the scoring in the 16th minute.
It took just four minutes of the second half for Leyds to extend his team’s advantage by chasing and grounding Dewaldt Duvenage’s grubber kick.
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Video: Dan Carter exclusive
Sefa Naivalu crossed unmarked on the right to score the Rebels’ first try of the match after the patient away team recycled possession through 18 phases.
And Naivalu bagged another in the 76th minute, going over down the right again after a series of pick and drives from the Rebels.
Stormers had the last word, though, Engelbrecht slicing through the opposition’s defensive line after some slick handling down the right to cap off the win with another try from the last passage of play in the match.
Video: Rebels coaches Wessels and Foote join the Late Tackle team