The Sixers batted first in the Big Bash League clash on Thursday night at the Gabba and made a brilliant start thanks to Michael Lumb, but then limped to just 8-140 from their 20 overs.
The Heat then were cruising at 3-95 after 14 overs needing only another 46 runs from 36 deliveries, but the home side capitulated through a combination of good bowling, a brilliant catch and disastrous late-inning batting.
Sydney speedster Brett Lee picked up two wickets in the 15th over dismissing Chris Lynn for 34 and then Craig Kieswetter for 38 when Silk took a stunning diving catch at mid-on.
Kieswetter struck his lofted drive sweetly and it appeared to be past the field and heading to the boundary, but Silk put in a full length dive and stuck out his left mitt, and the ball stayed in.
Not only was the catch itself a remarkable sight, but it also turned the match in the Sixers' favour.
Once Kieswetter left, the Heat were 5-97 and that brought Brisbane captain James Hopes and Chris Sabburg to the wicket, and that partnership won the game for the Sixers.
The pair were even booed by their own hometown fans for their reluctance to try and run for two and simply never looked like hitting anything close to a boundary in a 24-run partnership off 26 deliveries.
Those two batting together took the Heat from being in a comfortable position to putting themselves out of the game when Hopes was out for nine and Cutting came to the crease.
By the time Cutting got on strike with two balls to go, the Heat needed 11 runs to win and he hit the first ball off Josh Hazlewood sweetly over mid-off for six, and tried to do so again off the final ball but fell five metres short and was caught in the deep.
Even though Nic Maddinson fell for nine early in the Sixers' innings to start the match, they got off to a terrific start thanks to Englishman Lumb.
The left-handed opener set up the innings beautifully as he racked up a terrific 56 from 42 balls with six fours and a six before falling to Cutting when caught by Hopes in the 13th over with the Sixers' score on 3-95.
He had received some good support from Moises Henriques (19 from 15), Marcus North (13 from 16) and Ravi Bopara (13 from 11), but then things began to unravel for the Sixers.
The Heat's horror finish to their innings was highlighted by them having just two scoring shots in the last five overs going for more than a single.
No player in the last five at the crease reached double-figures as the Sixers limped to 8-140 from the 20 overs.
Cameron Gannon finished with 4-24 from four overs with the ball for the Heat while Cutting took 2-30 from four, Daniel Vettori 1-25 from four, Mark Steketee 1-15 from two, Hopes 0-26 from four and Dan Christian 0-18 from two.
Luke Pomersbach and Kieswetter didn’t set the world alight at the start of the Heat innings, but nor did they need to requiring just over a run a ball for victory.
The pair combined for a 30-run opening stand in 28 balls before Pomersbach was out for 15 when run out.
The Heat then lost 3-3 over the next two overs with Joe Burns out for one and Christian also for one, to Josh Lalor and Henriques respectively, but the home side then appeared to steady through Kieswetter and Lynn.
Lynn was looking to be the match-winner for Brisbane moving to 34 from 21 balls with two fours and two sixes, but Lee then came back on to bowl for the Sixers and things changed dramatically.
Lee bowled a full ball first up to Lynn and it bounced off his pad onto the stumps and then three balls later Kieswetter was out to the amazing catch from Silk.
The Heat then did not manage to hit a boundary for 38 balls until Cutting managed to do so on the second last ball of the innings, but it was too little too late for the home side.
Lee finished with 2-22 from four overs for the Sixers with Hazlewood taking 1-43 from four, Lalor 1-18 from four, Henriques 1-12 from two, Steve O'Keefe 0-24 from four and Bopara 0-11 from two.
The Big Bash League now continues on Friday night with the Perth Scorchers hosting the Sydney Thunder at the WACA Ground in Perth.