The Welsh side are unlikely to make it into the Amlin Challenge Cup as one of three pool runners-up who drop into the tournament, but finished their European campaign on a high in Scotland.
Dan Parks and Chris Paterson exchanged early penalties before Wales wing Leigh Halfpenny put the visitors in front with a long range penalty.
The Blues defence had held firm with repeated attacks coming their way from the hosts, but Ben Cairns got the try Edinburgh were looking for to take them into the break 11 v 6 ahead.
Paterson and Parks again kicked a penalty apiece before running rugby took over and the Blues enjoyed long periods of sustained pressure.
Halfpenny kicked four unanswered penalties to reduce the deficit, take the lead and then secure the victory.
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