In reply to the Heat's 7-175 on Saturday night, the Scorchers were looking wobbly at 3-54 in the eighth over in front of 53,866 fans at Optus Stadium.
But a knock of 53 off 32 balls from Turner helped turn the tide the Scorchers' way, with the skipper combining with Josh Inglis (26 off 22 balls) for a crucial 86-run stand.
The disastrous run out of Turner with 40 runs still needed put the game back on a knife's edge.
Connolly stepped up to the plate, cracking an unbeaten 25 off 11 balls in a heroic performance.
The 19-year-old was dropped on 19 by Josh Brown, who failed to snaffle the skied chance.
The Scorchers needed 10 runs to win off Michael Neser's final over.
This time it was Nick Hobson turn to step up to the plate.
Hobson, who was responsible for the run out of Turner, cracked a huge six and then followed it up with a four to secure victory with three balls to spare.
Scorchers players streamed onto the field in wild celebrations after the winning runs secured the franchise a record fifth title.
Hobson finished unbeaten on 18 off just seven balls.
The brave Heat did well just to make the decider.
The team sat in last spot on the ladder just 22 days ago before winning four of their last five games to sneak into the finals in fifth spot.
The Heat then won away finals against the Sydney Thunder, Melbourne Renegades and Sydney Sixers to make the decider, but they started as rank underdogs against a Scorchers outfit hunting their another crown.
Brisbane were even labelled "nobodies" by the local newspaper in Western Australia due to the absence of Usman Khawaja, Matt Renshaw, Mitchell Swepson and Marnus Labuschagne to Test duties.
"Just FYI. We keep our receipts," the Heat tweeted on Saturday alongside a picture of the liftout that was headlined: "Scorchers versus a bunch of nobodies".
Those 'nobodies' stepped up to the plate and gave the Scorchers an almighty scare before Turner, Connolly and Hobson got Perth home.
