Stoinis and South African Tristan Stubbs put on 70 in 41 balls to turn a match that was going the league leaders' way.
Brave were already eliminated from contention but Rockets needed a win to seal a place in the final. Now they need results to go their way to avoid the eliminator.
Put in to bat, Rockets made 5-158 thanks to a late flurry from New Zealand's Mitchell Santner (35 not out in 23 balls).
Stoinis took 1-29 from his 20 balls and Nikhil Chaudhary 1-20 from 15 but both were outshone by uncapped veteran Dan Worrell. Included as one of three changes - with Jofra Archer rested ahead of England's Test series with Pakistan - Melburnian Worrell took 1-21, bowling 20 of the first 30 balls.
None of Rockets' top order really got going, Kiwi Finn Allen with 23 off 12 doing best.
Tim David was among those to struggle. The Australian international admitted beforehand he had found six-hitting difficult in the Hundred where the wickets are less batter-friendly than the IPL or Big Bash and outfields larger than in Asia.
David went on to make 15 off 12 with one six before skying one off Adil Rashid.
England wicketkeeper Jamie Smith (26 off 18) gave Brave a strong start in reply but they were 3-78 off 52 when Stoinis came in, leaving them with 81 needed off 48 and the ball getting softer.
He smoked hit his first three balls, from much-touted young English legspinner Calvin Harrison, for six-four-six and Brave never looked back.
The West Australian looked set to take Brave to victory and complete a half-century only to cloth a full toss to wide mid-off. In came Chaudhary with 11 required off seven, but he was able to enjoy a ringside seat as Stubbs hit a six to win the match with three balls to spare.
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