The visitors got off to the worst possible start in southern Ireland as Danny Barnes crossed for a try for Munster after just three minutes. Ian Keatley added the extras for the Champions as they raced into an seven point lead.
Dan Biggar opened the Ospreys account with a penalty after 8 minutes, but that was cancelled out by Keatley a few minutes later who kicked three points of his own.
Keatley and Biggar then missed a penalty apiece before the Ospreys playmaker kicked his second score right on the stroke of half-time.
The Ospreys drew first blood in the second half with another Biggar penalty to bring the scores to within a point.
As the game approached the final quarter, Ospreys scrum-half Rhys Webb touched down for the crucial score which gave his side the lead for the first time in the game.
Keatley reduced the deficit to a solitary point five mintes later with another penalty but the Ospreys hung on and as the clock reached 80, Biggar cemented the win with another penalty.
