Chasing 8-160, the Hurricanes were cruising after some top work by Ben McDermott (59 from 36) and Beau Webster (51 from 43) before a horror collapse at the death.
Hobart needed only 25 with four overs to go on Wednesday night at Ninja Stadium but lost 4-6 in 13 deliveries.
They required six runs off the final over but Pakistan import Zaman Khan held his nerve, conceding just two runs and also getting rid of Nikhil Chaudhary.
A win would have secured top place on the ladder and a home final for the Hurricanes but they could now be overtaken by both the Perth Scorchers and Melbourne Stars and drop to third.
The win lifts the Heat from fifth to fourth on the ladder and keeps their finals hopes in their own hands with one match to play against the Sixers.
The Hurricanes finished at 8-157 after being 2-124 in the 15th over.
Heat quick Xavier Bartlett (3-44) played a big part, picking up two wickets and conceding just five runs in the penultimate over.
"It was an unbelievable game of cricket," Bartlett told FoxSports.
"I got a bit of tap early. But it was good to come back and get some wickets. We always think we can defend anything."
McDermott, who was earlier dropped twice in the outfield by Max Bryant, was bowled by man-of-the-match Matthew Kuhnemann (2-24) in the 16th over.
McDermott and Webster combined after the Hurricanes fell to 2-26 early in the chase.
Webster, who starred in the fifth Ashes Test with an unbeaten 70 and three wickets, earlier took the scalp of Usman Khawaja with his off-spin.
The Hurricanes could be sweating on the fitness of allrounder Chris Jordan who left the field after pulling up sore with a foot problem but still came out to bat.
After being sent in, the Heat were slow out of the blocks and didn't hit their first boundary until the 16th ball of the innings.
McSweeney (49 from 32) and Matthew Renshaw (37 from 25) got the Heat going through the middle overs but they fell away, losing 4-13 in the last three overs.
Riley Meredith (3-40), Rishad Hossain (2-27) and Nathan Ellis (2-30) shared the spoils with captain Ellis getting the big scalp of McSweeney on the first ball of the power surge.
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