With the Thunder chasing 141 for victory and needing five to win off the final delivery, Nicola Carey connected beautifully as she slog swept Jess Jonassen over deep square leg but Birkett made terrific ground to pluck a high catch to leave the Thunder heartbroken.
The Thunder's chase started poorly when Rachel Priest fell for a duck in the opening over to player of the match Sammy-Jo Johnson (1-12 off four overs).
The home side looked the goods while the dangerous Rachel Haynes, who hit two sixes in her 44 off 33 deliveries, was at the crease but when she holed out, the Heat, led by Johnson, completely strangled the Thunder in the middle overs.
Captain Alex Blackwell (29) also started sedately before teeing off on Delissa Kimmince late and ultimately holing out to Laura Wolvaardt off Jemma Barsby (3-23) in the penultimate over before Carey (19) came literally within centimetres of dragging the Thunder into the decider.
Earlier, Johnson (33) was her usual attacking self before Laura Harris (32no) produced a gem of a cameo at the end to lift the Heat to 7-140 on a tricky deck.
Opener Grace Harris spooned a Taylor full toss to Naomi Stalenberg before Beth Mooney, fresh off a match-winning century against the Thunder a week ago in Cairns, could manage only 14.
Skipper Kirby Short departed for a golden duck to leave the Heat wobbling at 3-38 before Johnson guided the Thunder to safety with bat and ball.
"To put 140 on the board, we thought was competitive," Johnson said.
"(Brisbane) have got a really good batting line-up so we knew we had to bowl well.
"Stoked we could pull it off."