Tom Marshall sparked the comeback with a line break setting up a Matt Todd try, the visitor's only try of the match, while Carter's replacement Tyler Bleyendaal slotted a brace of penalties and a conversion in final quarter of the first half.
Bleyendaal kicked two more penalties in the second half to finish with 14 points.
Joe Pietersen kicked three penalties for the home side.
The Crusaders' third win of the season, after losing their opening two games, sees them leapfrog the Blues and Hurricanes into second spot on the New Zealand Conference and fifth on the overall standings.
They end their trip to the Republic by taking on the South African Conference leading Sharks next weekend in Durban.
Crusaders 19
Try: Matt Todd. Penalties: Tyler Bleyendaal 4, Con: Bleyendaal
Stormers 14
Try: Siya Kolisi. Penalties: Joe Pietersen 3.
- AP/nzherald.co.nz