Stubbs and Marcus Stoinis were instrumental in pulling off the chase as they shared a 70-run stand for the fourth wicket.
Aussie all-rounder Stoinis whacked four fours and three sixes to make 46 off 25 balls before he miscued a full toss from Ben Sanderson in the penultimate set and was caught.
However, with eight needed off the final five balls, bowled by Lewis Gregory, Stubbs and Nikhil Chaudhary got Brave over the line.
Stubbs sealed the win by whacking Gregory over square leg for six as he finished unbeaten on 52 from 28 balls, having also struck four fours and three sixes at a strike rate of 185.71.
The Rockets had earlier posted what looked a competitive total with the top order all chipping in with useful contributions.
Mitchell Santner top-scored with an unbeaten 35 off 23 balls as he shared an unbroken sixth-wicket stand with Gregory (28 not out off 18 balls) worth 66 runs.
The wickets were shared between five bowlers, with Brave skipper Chris Jordan curtailing Finn Allen’s attacking innings on 23 with a superb yorker.
It was a chastening day for 17-year-old quick bowler Manny Lumsden, though, as his first set disappeared for 19 runs and he was not called upon again.
