Kwagga Smith scored two tries and Andries Coetzee, Courtnall Skosan, Aphiwe Dyantyi and Hacjivah Dayimani all also went over as the Lions moved into a playoff place with a much-needed bonus-point home win at Ellis Park on Saturday.
Johan du Toit, Herschel Jantjies and Joshua Stander replied for the visitors from Cape Town, whose own playoff hopes are now in peril.
The Lions, who dropped Springbok pivot Elton Jantjies for disciplinary reasons, had a narrow 13-10 lead at half-time but two quick tries after the break effectively broke the Stormers' resistance.
It was the first win in five attempts this season for the Lions over fellow South African opposition.