Six of the Treviso line-up, including Barbieri, played for Italy in March's 13-6 Six Nations reversal that handed Scotland the wooden spoon and it was a similar story at Murrayfield.
Edinburgh, who saw Tim Visser, Lee Jones and Dougie Fife go over to secure a losing bonus point, have now lost their last three RaboDirect PRO12 games and were under pressure for most of this contest.
Just 90 seconds were on the clock when Greig Laidlaw had his first shot at the post, eight metres in from touch, but the Scotland international's effort was wayward.
Two minutes later though he made amends and gave the hosts an early lead.
Alberto Di Bernardo, whose last-gasp penalty secured victory for Treviso over the Scarlets at the weekend,missed the chance to level the scores with a long-range effort of his own on ten minutes.
The Italians though were continuing to press Edinburgh back and got their reward on 28 minutes when referee Matteo Liperini awarded a penalty try for a repeated scrummage infringement, Di Bernardo making light work of the conversion.
Five minutes later in-form Visser wowed the Murrayfield faithful with a superb 50-metre dash, outpacing Luke McLean, to cross for his eighth try of the campaign.
Laidlaw's conversion affirmed Edinburgh's advantage but Di Bernardo slotted away a 38th-minute penalty to bring the scores level at 10-10.
And, with time ticking towards the 40-minute mark, the visitors edged in front again when Barbieri took advantage of Matt Scott's error to waltz over for his fifth try of the season.
Di Bernardo held his nerve and gave Franco Smith's troops a 17-10 half-time lead and started the second in the same vain with a brilliant three-pointer.
His chance to extend Treviso's lead to 13 points was wasted on 53 minutes when the Argentinian-born fly-half missed a difficult kick.
But it mattered little seconds later as the impressive Barbieri charged down Richie Rees' kick and touched down for his second of the game.
Jones gave the hosts immediate hope when he cut through the outside and went over for his first RaboDirect PRO12 try of the season.
The hosts continued to press and got their rewards on 78 minutes when replacement Fife went in under the posts but they couldn't find a last-gasp winner and had to settle for a losing bonus point.
|Edinburgh Rugby Score Card|
|Benetton Treviso Score Card|
|Alberto di Bernardo||3||2||12|