The home side took the lead when Ospreys scrum half Kahn Fotuali'i was penalised for dissent and Ronan O'Gara knocked over the penalty.
Matthew Morgan drew Ospreys level in the 15th minute when Stephen Archer was caught on the wrong side of the ruck for Munster.
In an evenly contested affair, the Ospreys edged in front in the 24th minute when Morgan slotted his second penalty.
Alun Wyn Jones was penalised at a ruck two minutes before the break to give O'Gara an opportunity to level the scores. The experienced campaigner was in no mood to miss the chance as he ensured the scores were locked 6-6 at the break with his second penalty success.
The hosts went in front four minutes after the break with a well-constructed try. Dave Kilcoyne caused the initial damage, punching holes in the Ospreys defence, The ball was worked more central to Damien Varley who burrowed over from close range to give Munster an 11-3 lead with his sixth try in the red shirt.
O'Gara had no problem with the conversion to stretch their advantage out to seven points, but a yellow card for tight-head prop Stephen Archer for failing to use his arms in the tackle against James King knocked Munster back on their heels.
The Ospreys hit back after Jonathan Thomas emerged from under the mass of bodies with his third Pro12 try of the season. Morgan nailed the difficult conversion from the left to level the game for the fourth time, but they otherwise failed to take advantage of their numerical advantage.