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.
Sydney Sixers beat Brisbane Heat in a super over to reach Saturday's Big Bash League final, where they will play Perth Scorchers.