Ronan O'Gara matched Matthew Morgan's two penalties for a 6-6 scoreline at the end of a first half full of endeavour and hard running.
The hosts then moved ahead thanks to Damien Varley's 44th-minute try, only for the Ospreys to respond with a converted effort from Jonathan Thomas.
Steve Tandy's men touched down while Munster prop Stephen Archer was in the sin-bin, but there were no further scoring chances in an energy-sapping final half hour.
Munster, and O'Gara in particular, got off to a solid start with the veteran number 10 clipping over a fourth-minute penalty after some back chat from Kahn Fotuali'i.
Returning Munster captain Doug Howlett was found on the right touchline by a pinpoint kick from O'Gara, but Richard Fussell succeeded in bundling the New Zealander into touch.
The Ospreys, shorn of eight Wales internationals, rallied behind skipper Alun Wyn Jones and he was brought down close to the line after a muscular maul drive.
The attack produced a 16th-minute penalty that Morgan comfortably slotted, punishing Archer for a ruck offence.
The Ospreys looked sharper in possession and at the breakdown. A second effective lineout maul led to another shot at the posts for Morgan and the 20-year-old confidently delivered from a right-sided position.
The defending champions looked lively on the counter attack, one such instance seeing Thomas offload brilliantly to Fotuali'i, who set up Fussell for a kick chase.
Jonathan Spratt soon pressed from an intercepted pass, but Munster were also coming alive. Denis Hurley latched on to O'Gara's clever kick to the left and Tom Habberfield had to chase back to deny Hurley in the corner.
Approaching half-time, Habberfield beat Hurley to an inviting kick from Fotuali'i before he was tackled into touch by the covering Felix Jones.
Munster managed to get back on level terms two minutes before the break, with O'Gara firing over from inside the 22.
Munster gained further momentum on the restart, going through a mammoth 20 phases before hooker Varley wrestled his way over for the opening try.
Casey Laulala threatened in the left corner before the forwards took over, edging closer through a series of pick and goes, and Sam Lewis and Thomas were unable to keep Varley out.
O'Gara's conversion put seven points between the sides, however prop Archer's subsequent sin-binning for tackling James King without the ball gave the Ospreys an opportunity to hit back.
The Welshmen turned down a kickable penalty and that proved to be the correct decision as number eight Thomas piled over on the end of a lineout maul.
Morgan followed up with a tremendous conversion from the left touchline and the game became rather listless heading into the final quarter, with both sides making errors on the ball.
A costly knock on by Laulala ruined a promising Munster move. The hosts had the better of possession late on, but some crunching tackles from man-of-the-match Fotuali'i, Ashley Beck and Dmitri Arhip kept them at bay.
In the final play, O'Gara was set up for a long range drop goal which tailed away to the right of the posts.
The result, on the back of a thumping defeat in Treviso, leaves Munster in sixth place with five rounds remaining. The Ospreys stay six points ahead of them and the Scarlets, who currently hold the last semi-final spot, are a further point in front.
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