Charlie Shiel touched down for the second time to snatch a 24-17 victory with the last play.
Glasgow Warriors-bound centre Robbie Nairn scored the Scots' other try.
After losing to eventual champions New Zealand, Scotland had beaten Ireland, Italy and Wales to reach the fifth-place play-off.
The victory in Georgia was the second by a Scottish rugby team over its Australian counterparts this weekend, with Gregor Townsend's senior side beating the Wallabies on Saturday.
U20s head coach John Dalziel, assisted at the championship by former Edinburgh flanker Roddy Grant, will now take charge of Scotland Sevens.