Moises Henriques hit 18 runs as the Sixers scored 22 in their extra over, while the Heat made 15 in reply.
Brendon McCullum had earlier scored 46 as the Heat compiled 167-9.
The Sixers looked well set after Henriques' 64, but Ben Dwarshuis had to score two runs off the final ball to tie the game at the Gabba.
Henriques then hit two sixes and a four in the super over.
The Sixers will play the Scorchers, who are two-time champions, in the final at the Waca in Perth.

Belgium's Gianni Meersman sprinted to victory, while Darwin Atapuma retained his overall lead despite being one of the riders caught behind the collision.
Colombian Atapuma lost more than a minute, but organisers applied the 3km rule to determine finishing times.
Froome and second-placed Alejandro Valverde were part of the bunch sprint having been ahead of the crash.
The accident split the pack in the closing stages and ended the race of Dutch rider Steven Kruijswijk, who sustained a suspected broken collarbone.
It was a second win in this year's Vuelta for Meersman, who held off the challenge of Italy's Fabio Felline and Kevin Reza of France.
Team Sky's Froome is aiming to become the first man in 38 years to win the Vuelta and Tour de France in the same season.
1. Gianni Meersman (Bel/Etixx-Quick-Step) 4hrs 16mins 41secs
2. Laurent Didier (Lux/Trek) same time
3. Kevin Reza (Fra/FDJ)
4. Luis Leon Sanchez (Spa/Astana)
5. Zico Waeytens (Bel/Giant)
6. Alejandro Valverde (Spa/Movistar)
7. Romain Hardy (Fra/Cofidis)
8. Jempy Druecker (Lux/BMC Racing)
9. Kenneth Vanbilsen (Bel/Cofidis)
10. Jose Goncalves (Por/Caja Rural)
1. Darwin Atapuma (Col/BMC) 17hrs 39mins 52secs
2. Alejandro Valverde (Spa/Movistar) +28secs
3 Chris Froome (GB/Team Sky) +32secs
4. Nairo Quintana (Col/Movistar) +38secs
5. Esteban Chaves (Col/Orica) same time
6. Gonzalez Sanchez (Spa/BMC) +1min 7secs
7. Andujar Fernandez (Spa/Movistar) +1min 10secs
8. Leopold Konig (Cze/Team Sky) +1min 12secs
9. Peter Kennaugh (GB/Team Sky) +1min 14secs
10. Gianluca Brambilla (Ita/Etixx-Quick-Step) 1min 22secs

Choupo-Moting, 28, is their fourth signing of the summer after Darren Fletcher, Kurt Zouma and Josh Tymon.
"Eric has been on our radar for several years and we actually tried to bring him to the club four years ago," chief executive Tony Scholes said.
Stoke open their Premier League season with a trip to Everton on Saturday.
Choupo-Moting, who has also played for Hamburg and Mainz, has made 200 Bundesliga appearances and played in the Champions League with Schalke.
A former Germany youth international, he switched international allegiance to Cameroon in 2010.
He represented the Indomitable Lions at the 2010 World Cup in South Africa and has won 48 caps, scoring 13 goals.
Find all the latest football transfers on our dedicated page.
Sydney Sixers beat Brisbane Heat in a super over to reach Saturday's Big Bash League final, where they will play Perth Scorchers.
[NEXT_CONCEPT]
Britain's Chris Froome remains third in the Vuelta a Espana after a late crash caused chaos on Wednesday's stage five.
[NEXT_CONCEPT]
Stoke City have signed Cameroon winger Eric Maxim Choupo-Moting on a three-year deal after his contract ran out at German club Schalke.