Finch held his nerve in a tense finish, hitting Doug Bracewell for six off the fourth ball of the final over, after the Australian side found themselves needing 48 off the last four overs.
Finch found a solid partner in Andrew McDonald, who hit 18 not out in an unbeaten stand of 66.
"We nearly left it a bit too late, but we got over the line," said Finch, who made his runs off 60 balls with 11 fours and three sixes.
In Image: Victorian Bushrangers Aaron Finch (R) and teammate Andrew McDonald celebrate winning by 9 wickets
Text: Agencies
Images: AFP