Set 119 in Adelaide, Perth began well through Elyse Villani (23) and England captain Charlotte Edwards (18).
But Brunt was trapped lbw for 18 from 20 balls in the last over as they ended on 110-9, losing by eight runs.
Hobart Hurricanes face Sydney Sixers on Friday for a place in Sunday's final.
Hurricanes skipper Heather Knight and Sixers all-rounder Laura Marsh are the only England players left in the competition, although Marsh is not in the Sixers' 13-strong squad for Friday.
Thunder - the only Women's Big Bash team without any England internationals - were indebted to captain Alex Blackwell (39) and West Indies opener Stafanie Taylor (24), who helped them post 118-6 in 20 overs.
Thunder's Australia seamer Rene Farrell held her nerve to take two wickets in the final over of the match - Brunt and ex-England all-rounder Nicky Shaw - to finish with 3-20.
Meanwhile, Thunder's male counterparts also reached the final of the men's Big Bash League, beating Adelaide Strikers by eight wickets.

It was the first time the Single Transferable Vote (STV) system had been used to select two members in the same ward in a by-election.
The SNP topped the vote in the Leith Walk by-election, while Scottish Labour won the second seat from the Greens.
The by-election was called after Deidre Brock of the SNP and Maggie Chapman of the Scottish Greens stood down.
The SNP's John Lewis Ritchie topped the Leith Walk poll with 2,290 votes. He was elected at stage one in the STV process with a swing in first-preference votes of 7.6% from Labour.
Labour's Marion Donaldson received 1,623 votes, ahead of Susan Jane Rae of the Scottish Greens on 1,381.
Ms Donaldson was elected at stage 10 of the voting process after other preferences had been considered.
The by-election was called after Ms Brock stood down when she was elected as the SNP MP for Edinburgh North and Leith in May.
Ms Chapman, of the Scottish Greens, resigned from her post to concentrate on standing for the Scottish Parliament in next May's election.
The turnout for the by-election was 25.1%.
The SNP also held the Midlothian West seat on Midlothian Council with a swing of 6.3% from Labour.
The party's Kelly Parry secured 1,540 votes, ahead of Labour's Ian Miller on 945 votes.
The by-election was called after Owen Thompson was elected as SNP MP for the Midlothian constituency.
A late rally from England's Katherine Brunt could not prevent Sydney Thunder from beating Perth Scorchers to reach the final of the inaugural Women's Big Bash League.
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Two new councillors have been elected in a by-election in the City of Edinburgh.