Matthew Morgan scored all of the Welsh region's points in the first half as they took a 13-8 lead into the break, with the Scottish team's response coming through WP Nel's try and a Gregor Hunter penalty.
But the hosts dominated the second period and five more Hunter three-pointers gained a crucial win.
Although the Ospreys started the season in disappointing fashion, losing their first three matches in the Pro12, they have bounced back superbly.
Five consecutive victories leading into this encounter put them back in play-off contention. However, with both teams missing a number of internationals it was no surprise to see the game start in scrappy fashion.
Morgan's interception was the only action of note in the opening exchanges before the visiting stand-off edged them ahead after 16 minutes. Hunter levelled the scores soon after with a penalty, but the Welsh region's fly-half second from the tee regained their advantage.
Despite controlling the majority of the first half, the Ospreys were punished when the hosts' first real try-scoring opportunity ended in Nel touching down. Dougie Fife made the initial break before good hands from Richie Rees and Hunter sent the prop over.
Edinburgh's lead didn't last for long as Morgan crossed the whitewash after a spell of pressure to give them a deserved 13-8 lead at the interval.
Visiting prop Owen Williams was sin-binned at the start of the second period and Hunter added two three-pointers during that time to take the Scottish outfit ahead.
Further penalties from the home side's fly-half extended that advantage to 20-13. And Hunter sealed the win with a last minute kick from the tee.
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