The Gunners failed to fire from the start and it looked like it would be the reigning Celtic champions night as Stefan Terblanche scored for the Welsh side after barely a minute.
Matthew Jones added the conversion but he could not extend the lead with two missed penalties letting the Gunners of the hook for their misdemeanors.
Edinburgh got their act together midway through the half and their pack started to make inroads through prop Allan Jacobsen and Taylor. A solid maul saw the Gunners claim their first try when Al Dickinson reached the line from close range.
Chris Paterson added the extras with a confidence-boosting kick from out wide.
The hosts then snatched the lead through a Paterson penalty, but the Ospreys hit back through a penalty of their own as Jones punished the Gunners for a tackle offence.
Edinburgh started the second half full of fire and were soon rewarded when Scott Murray stole a lineout and after some slick work, Taylor went over to give Paterson a simple conversion. .
The Ospreys bounced back from the restart as speedster Johnny Vaughton outpaced the Edinburgh defence, before sending Jones over in the corner.
Rob Dewey then scored a wonderful solo try after smashing his way through two defenders from the halfway, and Paterson's third conversion opened up a nine-point gap.
Jones scored a last-gasp penalty for Ospreys, but it was the hosts who held on for victory.
|Edinburgh Gunners Score Card|
|Ospreys Score Card|