Shakib???s team-mate, legspinner Fawad Ahmed (1-7), also bowled the most economical four-over bowling spell in the history of the Big Bash, eclipsing Daniel Vettori???s 2-10, as the Heat were bowled out for 80 in 17.3 overs.
Matthew Wade fell second ball on the Renegades run-chase ??? Andrew Flintoff taking a terrific leaping catch at slip ??? to James Hopes, who had his own impressive return on 1-8 from his four overs.
Ben Stokes (4) followed soon after, leaving the crease in disbelief as Dan Christian (2-13) clung on to an instinctive return catch in his follow through as a firm drive arrowed in at his head.
They limped along to 39-5 when Shakib fell in the 10th over to a poor shot given the match situation, upper-cutting a Christian ball straight down the throat of third man and it looked like a tense finish was on the cards.
But Tom Beaton played the crucial match-winning knock, with 31 off 16 balls, including two sixes, as the Renegades ended up winning comfortably with six overs to spare.
He was well supported by the out-of-form Ben Rohrer (25no) who picked the best time to make his first double-figures score of the tournament.
Earlier James Pattinson had given the Renegades the perfect start with the ball, taking two wickets with the first two balls of the match as Jimmy Peirson drove to point and Joe Burns was bowled by one that came back off the seam to hit off-stump.
Christian defended the hat-trick ball and struck Pattinson for six but was then stumped by Wade off Shakib.
The spinner also bowled Jason Floros through the gate with the very next ball and while Chris Lynn survived the hat-trick ball, soon after he gifted a simple return catch to register the fourth duck of the innings.
Peter Forrest (22), played a lone hand until Alex Doolan grabbed sharp chance at slip at the second attempt off Fawad, to make it 36-6 in the eighth over.
Flintoff???s hopes of rescuing the Heat came to an abrupt end when he was called through for a single by Ben Cutting, only to be left stranded mid-pitch as Shakib quickly collected the ball to run him out for three at the non-striker???s end.
Cutting made what amends he could, top-scoring with 30 off 35 balls before he was bowled by Shakib, his fourth wicket.
But the Renegades chased down the target to still give them a hope of qualifying for the semi-finals.