The sides went into the break all square at half-time, and scored a try apiece early in the second half through Damien Varley for Munster and Jonathan Thomas for the visitors. But despite playing for another 30 minutes, neither side could break the deadlock.
Munster, who had suffered defeat in three of their last four RaboDirect PRO12 games including a loss to Treviso last weekend, started on the front foot and took an early lead.
Ronan O'Gara slotted a fourth-minute penalty to put Munster 3-0 ahead but the home side were unable to add to that lead and were pegged back on 16 minutes.
Ospreys fly-half Matthew Morgan coolly knocked over a penalty of his own to bring his side level after Munster were penalised at the lineout.
And nine minutes later Morgan kicked the Ospreys - who came into the game missing 28 players through injury or international duty - into the lead with his second penalty of the night.
And it stayed that way until the latter stages of the first half despite some bullocking runs from Dave Kilcoyne and Stephen Archer that kept the Ospreys on the back foot.
Two minutes before the break, O'Gara levelled the scores with his second penalty of the night to send the sides in all-square at the interval.
Munster roared out of the blocks in the second period and re-took the lead when they won a lineout on the Ospreys five-metre line.
The ball was worked through the forwards who battered against a resolute Ospreys defence until finally Damien Varley - who started the game ahead of rival hooker Mike Sherry - managed to squirm over the line.
O'Gara's conversion made it 13-6 on 44 minutes, but there cause was not helped by Archer's yellow card shortly after.
Referee Nigel Davies awarded the Welsh outfit a penalty deep in the Munster 22 but instead of going for three points Morgan kicked to the corner.
And from the set-piece the Ospreys made the extra man count, driving over wide on the left with Jonathan Thomas the man to touch down on 51 minutes.
It left Morgan with a tricky conversion from out on the touchline but the 20-year-old fly-half held his nerve to slot the ball home from a narrow angle.
That was it as far as the scoring was concerned, despite Ashley Beck impressing in the loose for the Ospreys with his power and offloading.
As the seconds ticked away Munster engineered field position in front of the posts but from distance O'Gara failed with his drop goal attempt, the ball fading badly to the right as the sides settled for the draw.
|Munster Rugby Score Card|
|Ospreys Score Card|