Jonny Vaughton and Steve Tandy went over in the first half before Stefan Terblanche, Justin Marshall and Huw Bennett went over in the second to secure maximum points in Scotland.
The Ospreys had to recover from going 7-0 down after twenty-four minutes when Matt Mustchin peeled off the back of a line-out out drive to break a resilient defence.
After a shaky start, the Ospreys conjured two tries against the run of play with some breathtaking rugby to lead 10-7 at the break. Jonathan Thomas stole an Edinburgh line-out and the Ospreys shipped the ball out wide at pace for Vaughton to scamper over before Tandy finished off a length of the field attack inspired by Andrew Bishop, Marshall and makeshift fly-half Jason Spice.
With Gavin Henson, James Hook, Shaun Connor and Matthew Jones all out, the Ospreys had to rely on tries to get their points and they delivered. Terblanche glided over after the Ospreys stole another Edinburgh line-out before Andy Lloyd's pin-point pass to Marshall saw the All Black scrum half scorch through for the bonus point.
Edinburgh prop Allan Jacobsen was sin-binned two minutes from time for stamping and the Ospreys made their advantage count, driving replacement hooker Huw Bennett over for the final score.
Edinburgh 12 Ospreys 30