Two tries from Jeff Hassler were not enough for the Ospreys as they were beaten 31-25 in Edinburgh.
The visitors had an early lead after the first of Hassler's tries.
But the home side responded, as Grayson Hart crossed the line twice, with Sam Beard also adding five points to Edinburgh's tally.
The Ospreys almost managed a remarkable second half comeback, after Hassler's second try put them within three points.
The Welsh side did manage a losing bonus point however towards their push for a play-off place.