A third-minute score from Harold Vorster helped the Bulls into a 7-3 lead at half-time but Evan Roos replied early in the second period.
Andre-Hugo Venter then added a second try for the hosts shortly after Bulls centre Cornal Hendricks was sin-binned.
A late drop-goal from fly-half Manie Libbok sealed the result.
The competition was previously known as the Pro14 but was rebranded for the 2021-22 campaign following the introduction of four sides from South Africa.
The result marked an 11th win in a row for the Stormers, who never won the title during their time in Super Rugby - and were beaten by the Bulls in their only final in 2010.