Tries from Niall Scannell and James Cronin saw Munster close the gap on the Conference A leaders, while three Tyler Bleyendaal penalties saw the home side pull clear as Harley earned a record 178th cap for Glasgow.
George Horne impressed for Warriors and bagged their only try of the game seven minutes into the second half, while the return of Richie Vernon was a major boost ahead of the season run-in for Glasgow.
Bleyendaal kicked the game’s first points after ten minutes and Munster stretched their lead 15 minutes from the break when Scannell scored off the back of a maul.
After Bleyendaal converted for a 10-0 after 25 minutes, Alex Wootton denied Matt Smith a certain try with a brilliant tackle, and moments later Bleyendaal slotted a penalty for a 13-0 half-time lead.
Adam Hastings cut Munster’s lead with a 43rd minute penalty for Warriors after a smart break from Brandon Thompson, and five minutes later Horne got the try his display deserved when he appeared on Lee Jones’ shoulder to score. Hastings converted to make it 13-10.
But the sell-out home support was lifted two minutes later when Cronin dived on his own kick through to restore Munster’s advantage.
Bleyendaal made sure of the win with a penalty 18 minutes from time.