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The visitors had taken an early lead through a Shaun Connor drop goal but it was Edinburgh who took the first try scoring opportunity with Simon Webster completing the superb touchdown following a individual run from half way.
However The Ospreys simply turned up the pressure and were rewarded by a Matt Jones penalty which retook the lead.
Jones was on the mark again four minutes later to add the conversion to a try by lock Andy Newman, who galloped through from 10 metres after a well-worked lineout move to put the Ospreys well ahead at the break.
A Phil Goodman penalty reduced the gap just after the break but smartly taken drop goals from Jones kept the Ospreys in command.
The league leaders then made sure of their victory when Jones slotted a further penalty before converting their second try from Stefan Terblanche.
Aled Brew bagged a third Ospreys touchdown but it was Edinburgh who claimed the last word with a Joiner try, goaled by Godman.