Set a huge 192 to win, Australia collapsed to 62 for four.
But, initially through Cameron White, who made 43, and Hussey's unbeaten 60, they recovered and will now face old rivals England in Sunday's final at Barbados's Kensington Oval.
Australia needed 18 to win off the last over, from off-spinner Saeed Ajmal.
Mitchell Johnson took a single off the first ball and then Hussey took over.
The left-hander pulled Ajmal for six, struck him over long-on for six more, brought the scores level with a four and thumped another six for good measure as Australia finished on 197 for seven.
The man-of-the-match faced just 24 balls, but hit six sixes and three fours.
In Image: Australia's Michael Hussey celebrates after hitting the winning shot
Scores | Coverage: ICCT20