It looked as if the match was heading for a 19-19 draw until the Ospreys counter-attacked in injury time.
These two team went into the game fairly evenly matched, the Ospreys were minus eight World Cup stars to Edinburgh's seven.
Against the run of early play Edinburgh scored a try on six minutes through wing Tim Visser after Richard Fussell over-ran Greig Laidlaw's chip ahead.
The Ospreys hit back with a stunning try from in-form scrum-half Rhys Webb, who weaved his way over from a line-out on the 22.
From the restart Edinburgh regained the lead with a Laidlaw kick but Dan Biggar locked the scores at 10-10 with a penalty from in front of the posts after 21 minutes.
The Ospreys saw points go begging in the space of two minutes.
First Biggar thought he had put over a drop goal but the referee ruled it out and then captain Tipuric spilled the ball in the act of scoring.
But the Ospreys went into the lead after 31 minutes when Biggar slotted a penalty from a
maul infringement by flanker Sean Cox.
However, there was a frustrating end to the first half for the Ospreys.
First the referee got in the way of an encouraging attack, then Kristian Phillips attempted a rather over-ambitious drop-kick before Laidlaw levelled things with a penalty on the final whistle.
Edinburgh conceded a penalty just 51 seconds into the second half, but Biggar's shot at goal rebounded off the left upright.
By contrast Laidlaw kicked his effort after 45 minutes to give Edinburgh a 16-13 lead.
A minute later Biggar had found the target to tie up the scores at 16-16 with his third penalty.
The Liberty Stadium crowd got very frustrated with referee David Wilkinson who seemed to give everything the Scottish region's way, twice blowing up for crossing offences when the Ospreys were building encouraging attacks.
The Ospreys were well down on the penalty count but they got one back in their favour when Edinburgh failed to roll away. Biggar kicked it to give his side a 19-16 lead after 65 minutes.
But Laidlaw levelled the scores after Ian Evans was sin-binned from a line-out offence.
The match seemed destined for a draw until Tipuric squeezed over in the second minute of injury time.
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