Dougie Fife and Lee Jones crossed in the second half for Edinburgh - to add to a first-half score from Tim Visser - however it wasn't enough with Barbieri bagging his second score for Treviso.
The Italian flanker took his try tally for the season to an impressive six while bagging his side their third win of what has been an impressive RaboDirect PRO12 campaign to date.
However despite the visitors' heroics Bradley was more concerned with the second-half errors of his side.
"Ultimately we are masters of our own destiny," he said.
"If we had approached our opportunities in the second half with more composure, we could have won the game.
"We've got a very big game next week and it's now become even more important because of (the Treviso) result."
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