Chasing 165, Brisbane were struggling at 4-65 in the 12th over, but Harris blasted nine fours and three sixes in a spectacular unbeaten 68 off 29 balls.
Harris, who had scored just 42 from her seven previous innings this season, and New Zealand international allrounder Amelia Kerr (41 off 33), pounded 87 in six-and-a-half overs of mayhem.
Kerr was dismissed with 13 runs needed off as many balls and though Brisbane lost a wicket with the scores level, Harris swept the winning boundary with two balls to spare.
Harris reached her 50 off just 18 balls, the second quickest half-century in WBBL history
"It was nice to score a few runs and contribute a bit more to the team, it was a slow start from me to the season," Harris told Fox Sports.
The win lifted Brisbane to second though the two teams immediately below them, Perth and Adelaide, were scheduled to play in the final game on Sunday, with the winner set to regain that lofty spot.
It was a devastating loss for fifth-placed Hobart, who remained three points off fourth spot, though they have played less games than the teams above them.
The two innings followed a similar pattern with both struggling in the first half and cutting loose in the second.
The Hurricanes took 13 off the first over, but lost three wickets in as many overs to crash to 3-20 in the fourth.
They hit just two fours and a six in the first 14 overs, but blasted 79 runs off the last six.
South African batter Mignon du Preez (73 not out off 54) paced the recovery perfectly, running hard early in her innings and clouting several boundaries in the closing overs.
She added 61 off seven overs with Nicola Carey (30 off 23), while Nicola Hancock (4-32) was the best Heat bowler.
The Hurricanes spinners chewed through the top order with Molly Strano (2-17) and Maisy Gibson (2-21) excelling.
The Heat went almost nine overs without a boundary before Harris smashed 14 off the last three balls of the 12th over to break the shackles.
Carey spilled a tough chance at deep midwicket when Harris was on 16 and the powerful batter made Hobart pay.
