Mike Blair, who featured in Scotland's winless RBS 6 Nations campaign this year, scored Edinburgh's only try while international teammate Greig Laidlaw kicked 14 all-important points in the win.
The triumph ensured Edinburgh became the first Scottish team to ever reach the Heineken Cup semi-finals where they will face Ulster after they overcame Munster 22-16 at Thomond Park.
Edinburgh don't have fond memories of the Irish outfit this season - losing 42-20 and 38-16 in their two Celtic League encounters in 2012 - but Bradley is hopeful of it being third time lucky later this month.
"Ulster went into the match with Munster as the form side and I think that got them through it and was a big part of a great start, which saw them take a19-point lead in half an hour," said Bradley told the Edinburgh Rugby website.
"True to Munster and the legacy they have built up over the years, they came back but into it but in the end it was just too much to claw back.
"We're really looking forward to playing Ulster in the semi-finals. We have played them twice already this season and come off worse on both occasions.
"However, a one off semi-final in the Heineken Cup is a different matter. We can take a lot of confidence both from our form on the road in this tournament and especially from our win over Toulouse at home.
"It is a match that we look forward to and one in which we are confident that if we do our job properly, individually and collectively we can get a win this match."