They had to dig deep in the first half when a robust Munster were leading after 24 minutes and apparently hanging on.
However, the home side dug in and in the second half were well on top and deservedly notched up the bonus-point win by running rampant late on.
Munster's first-half tries from Casey Laulala and David Kilcoyne both converted by Ronan O'Gara were countered by interception scores from Sean Maitland and Niko Manawatu, while Ruaridh Jackson converted both and kicked two late penalties that ensured Glasgow were ahead at half-time.
There was no stopping Glasgow after the interval with Jackson, Mark Bennett, Stuart Hogg and John Barclay all touching down with the former, who also kicked a penalty, converting the first pair and Duncan Weir the remaining brace.
Doug Howlett scored Munster's late consolation try converted by Ian Keatley and O'Gara kicked a penalty.
Glasgow were given the perfect opportunity within the first 30 seconds when their first man to touch the ball, Sean Maitland, intercepted a floated pass by Laulala to run in a try converted by Jackson.
But Laulala would make amends by scoring under the posts after number eight James Coughlan's barge had set up the pressure before O'Gara converted.
Storming Munster play put them into the lead in the 24th minute. Following a couple of driven line outs prop Kilcoyne got over from close range with O'Gara converting.
However within two minutes Glasgow were even thanks to another interception, Glasgow scrum-half Manawatu picking off from his opposite number Duncan Williams to even up the scores.
Glasgow then had the better of things just before half-time and were able to go in justifiably ahead thanks to penalties from Jackson in the 34th and 38th minutes.
Glasgow made a bright start after the break but Munster narrowed the gap to four points following a long penalty from O'Gara.
But after 58 minutes, the Warriors stuck together a flowing move started by Barclay and notably involving Pat McArthur, Moray Low and Josh Strauss before Jackson finished off in style and converted.
Now on top Glasgow extended their lead in the 66 minute with Jackson's third penalty.
The fourth try was coming and it arrived in the 74th minute. After a long break by DTH Van Der Merwe, young substitute Mark Bennett got over by the posts for Jackson to convert.
In the closing moments Glasgow took their try count to six through Hogg and Barclay with Weir converting both while Howlett had a consolation for Munster converted by Keatley.
Munster's chances of making the top four have taken a hit following this defeat. With their two closing matches against the two bottom sides, their next RaboDirect clash at home to Leinster becomes key.
But before that, they have the little matter next week of a Heineken Cup meeting with Harlequins.
Scotland star Greig Laidlaw booted Edinburgh to a narrow 9-7 victory over Zebre in Parma - but it could easily have been a miserable night for the visitors.
With time running out at the Stadio XXV Aprile, Zebre fly-half Luciano Orquera hit the crossbar with a drop-goal attempt, when had the ball landed the right side it would have been a near-certain match-winner.
Zebre remain without a victory in the RaboDirect Pro12, but this was a night when they went particularly close, scoring the game's only try when Dries van Schalkwyk crossed in the first half, and leading 7-3. Edinburgh cut the gap to a single point at the break, and Laidlaw kicked his third penalty of the contest five minutes into the second half.
That gave Edinburgh a lead they clung to, but it never looked as vulnerable as when Italy fly-half Orquera took aim with three minutes to go and so nearly claimed the spoils.
Zebre went into the match on the back of 18 consecutive defeats in the competition this season, and finished it with the small consolation of their eighth losing bonus point.
Laidlaw edged Edinburgh ahead with his first penalty in the third minute, but the lead was relinquished when South African Van Schalkwyk barged over in the 30th minute. Orquera converted.
Laidlaw reduced the arrears with the interval approaching and made it three successful kicks in three attempts soon after half-time.
Despite applying plenty of pressure in the second half, Edinburgh could not add to their two-point lead, with Laidlaw among those who failed to make the most of try-scoring opportunities.
Orquera almost punished them, but the visitors rode their luck to snatch their sixth win in the Pro12.