Fly-half Ronan O'Gara also stamped his mark on the game, kicking 23 points which included four second half penalty goals.
Glasgow, 22-9 winners when the sides met here twelve months ago, had edged their way to a 26-17 half-time lead.
Sean Lineen's men grabbed tries through winger DTH van der Merwe and scrum half Colin Gregor, with 19-year-old fly-half Duncan Weir kicking four penalties and two conversions.
Weir, who was making his first league start, added a penalty in a second half which the Munster pack dominated.
The key moments came just after the restart when Leamy set up Ronan for his third try of the campaign and then Murphy completed his brace, benefiting from a clever kick from O'Gara.
Moray Low returned at tighthead prop for the Warriors, in one of three changes Glasgow coach Lineen made to the starting line-up.
Weir got the nod for the out-half spot over Ruaridh Jackson, while Argentinian winger Federico Aramburu took over from Scotland Sevens squad member Hefin O'Hare.
Munster welcomed back three frontliners from injury - centre Lifeimi Mafi and flanker David Wallace, along with number 8 Leamy, who had recovered from the rib injury that forced him to miss last weekend's win over the Ospreys.
Weir missed two early penalty attempts, but settled after he scored at the third attempt. After O'Gara and his young rival missed a penalty apiece, Glasgow made the breakthrough for the first try, 16 minutes in.
Warriors captain John Barclay did well to retrieve possession from the restart and from a ruck position, Weir dangled a lovely kick over to the far wing where Bernardo Stortoni brilliantly knocked the ball back for van der Merwe to dive over for a seven-pointer.
Munster responded just four minutes later, Leamy crossing close to the posts after a lineout drive and good work from prop Tony Buckley. O'Gara's conversion reduced the arrears to 10-7.
Weir and O'Gara then traded penalties, with the former's a terrific strike from near the touchline. The Scotland Under-20 starlet banged another long range effort over, but that good work was undone by Murphy's 30th-minute try.
The former Leicester man, making his first appearance on the wing for Munster, had the gas to finish off a try, linking well with full-back Paul Warwick. O'Gara's conversion moved Munster ahead.
Their lead was a brief one, a ruck offence from Munster lock Donncha O'Callaghan landed him in the sin-bin and scrum half Gregor took advantage to notch Glasgow's second try. He broke away from the back of a scrum and scrambled through for a score which Weir converted.
It could have been more for the Warriors, number 8 Richie Vernon was denied a try in the corner by a terrific tackle from Munster centre Sam Tuitupou.
As they did against the Ospreys, Munster came out for the second half and grabbed a try off an early set of attacks. They cleaned up their discipline considerably and punished any Glasgow mistakes.
A Gregor knock on invited Munster forward and from a close-in scrum, Leamy broke and popped the ball to Ronan for a well-taken try, to which O'Gara added the extras.
The Munster forwards marched back into the Glasgow 22 and just four minutes after Ronan's effort, Murphy was put over in the corner by O'Gara's well-weighted grubber. That was the bonus point score and O'Gara kicked the conversion for a 31-26 turnaround.
Tony McGahan's charges tightened their grip on the game as O'Gara nailed four more penalties to one from Weir, who is 14 years his junior.
That greater experience and the confidence which comes from three opening wins told in the end as Munster, containing the Warriors' late rally, stood firm to claim only their second victory in their last six visits to Glasgow.
Speaking afterwards, Glasgow boss Sean Lineen said: "I was delighted with the first half. Some of the rugby we played was outstanding.
"We then did a couple of silly things at the start of the second half and we will need to cut out these errors ahead of next week.
"There will be a bit of pain with all the changes to the squad, so it's going to take time. Munster are singing in the changing room because it means a lot to win at Firhill."
|Glasgow Warriors Score Card|
|DTH van der Merwe||1||5|
|Munster Rugby Score Card|