The province had lost on their last three visits to the Swansea, so this was an important result for Rob Penney's men especially with Leinster pressing them for top spot.
Munster went for the jugular right from the off as they turned down a kickable penalty and created the opening try for flanker Sean Dougall in the sixth minute - direct from a close-in lineout maul.
Duncan Casey threw in to Donnacha Ryan who tapped down to captain James Coughlan, setting up a powerful drive that the Ospreys were unable to halt and the well-supported Dougall made the line.
Ian Keatley converted and although Fijian winger Aisea Natoga was a lively figure in the Ospreys' back-line, Munster's strength up front kept them on course.
The visitors moved 10-0 clear after a Keatley penalty rewarded a strong scrum from the pack, and he then cancelled out a successful Sam Davies place-kick to maintain the 10-point cushion.
The Ospreys, who were 75-7 winners over Benetton Treviso last time out, came more into the game approaching half-time.
Full-back Davies hit an upright with a 35th minute penalty attempt, but the home side registered their only try with a minute remaining.
Wales centre Ashley Beck broke a tackle from Keatley in the Munster 22 and had the pace and footwork to reach the line, touching down to the right of the posts. Davies missed the conversion though and dropped a long range penalty short with the final kick of the half.
13-8 ahead at the break, the Munstermen soaked up some early pressure before Simon Zebo ran in a real sucker puncher score from his own half.
The Ireland international seized upon a loose ball that squirted out of a ruck, spun out of a tackle from Tom Habberfield and dashed away over halfway and up the left touchline for his fifth try in four games.
Keatley missed the conversion and CJ Stander and JJ Hanrahan entered the fray for Munster, replacing Tommy O'Donnell and Keatley respectively, before Davies replied with a 68th minute penalty.
The Ospreys went close to a second try when hooker Scott Baldwin scrambled his way over. Munster breathed a sigh of relief though when television match official Tim Hayes ruled that it was a double movement.
Economical Munster made certain of the win in the 71st minute and their final try of the evening came again from a lineout platform.
Hanrahan charged through the middle off a Duncan Williams pass, linking with supporting James Downey who dummied and then used his strength to go over under the posts for his first ever Munster try, with Natoga hanging off him.
Hanrahan's conversion put 14 points between the sides and the Ospreys, who dominated the second period in terms of territory and possession, finished with 14 men after replacement Ryan Jones was sin-binned late on.