Edinburgh started with a bang with a try from Matt Scott but Blues centre Cory Allen and Filo Paulo gave the visitors a 17-10 lead at the break.
After the interval Dan Fish raced over before Tom Brown put the hosts back in the game by dotting down in the far corner with less than an hour played.
The match was eventually secured when influential Allen provided Dafydd Hewitt with the try bonus point in the corner from a well worked move.
Edinburgh were desperate to salvage a bonus point and did so with a 79th-minute try by flanker Cornell du Preez.
