The Heat's finals prospects appeared doomed when posting just 7-129 on the batsman-friendly Adelaide Oval.
But Brisbane then bowled the Strikers out for a paltry 97 in 18.2 overs.
Brisbane quick Cameron Gannon took three wickets and captain James Hopes, Ryan Harris and Daniel Vettori each claimed two victims to ensure the Heat remain in contention for the play-offs.
Brisbane play Hobart Hurricanes on Thursday with the winner likely to join Melbourne Stars, Perth Scorchers and Sydney Sixers in the finals.
Brisbane's trump card on Saturday was spinner Vettori, who with his Adelaide counterpart Johan Botha joined the list of most miserly BBL bowlers.
Vettori took 2-10 from his four overs - by economy rate, the third best bowling figures in BBL history.
His game-turning spell came after Adelaide captain Botha took 2-11 from his four overs - the fourth-best figures by economy rate, until taken down a rung when bettered by Vettori.
New Zealander Vettori and South African Botha produced their feats on an Adelaide Oval deck which was receptive to spin, but hardly a raging turner.
Both used guile rather than grip, with Vettori shrewdly employing drift and arm-balls while Botha favoured a flatter trajectory and doosras.
Vettori helped cripple Adelaide's run chase, with only Nathan Reardon (33), Tim Ludeman (22) and Michael Klinger (17) reaching double figures.
And earlier, only four Brisbane batsmen managed double-figures scores - a list headed Craig Kieswetter's 51 from 46 balls.
The Heat's 19-year-old debutant Ben McDermott, the son of Australian Test great Craig McDermott, made 30 from 26 deliveries and Luke Pomersbach (20) and James Hopes (12 not out) were the only other batsmen to pass nine runs.
Brisbane skipper Hopes described Vettori, who is still troubled by a back injury, as an "absolute freak".
"He can still barely walk," Hopes said.
"The ability to bowl that slow, with that control, and still get those amounts of revolutions on the ball - it's a pretty rare thing.
"He has shown over 15 years that he's probably one of the premier Twenty20 bowlers in the world."
Adelaide captain Botha couldn't hide his disappointment, with the Strikers now failing to make the finals in all three BBL tournaments despite their talented playing list.
"I don't have the answers," Botha said.