The visitors enjoyed the better of the opening exchanges and were six points up thanks to two Calvin Howarth penalties before Paul Burke opened Munster's account with a 12th minute penalty.
Things improved for Glasgow when Dan Turner crashed over for a try that Howarth converted and he added a penalty seven minutes later to put his side 16-3 up.
Munster responded with a Jason Holland try to leave themselves trailing 11-8 at the break.
In the second half Munster dominated and Burke narrowed the defisit with a penalty after just two minutes. Alan Quinlan tied the match when he drove over from a line-out maul and Burke's conversion edged them in front.
Rob Hnederson marked his introduction with an incisive break and his clever inside pass found James Storey who scored under the posts.
Try as they might Munster were unable to get the fourth try that would have given them the bonus point, in the end The Scots earning themselves what could turn out to be a crucial return from their first visit to Thomond Park.