Munster broke the deadlock in the fifth minute with a well-struck Mossie Lawlor penalty when Glasgow sneaked offside 35 metres out.
They then stunned the home fans with two tries which were dubious to say the least.
First, Davies ignored home pleas for a knock-on and allowed winger John Kelly to get the vital touch to a clever cross-kick by Jason Holland.
Then after soaking up a series of Glasgow raids, they took full advantage of the official failing to spot a blatant offside in midfield.
Alan Bulloch was robbed on the Munster 22-line and Martin McPhail cruised in for the breakaway score in the wake of slick interplay with Eoin Reddan.
Seconds earlier, Glasgow had wasted a great opportunity following a high-speed build-up featuring Glenn Metcalfe, Calvin Howarth and Sam Pinder.
The home side suffered a setback when Howarth limped off to be replaced by Stuart Moffat.
And they were forced to show off their defensive talents in the lead-up to the break as the Irishmen piled on more pressure.
Munster kept up the momentum after the restart and only some committed tackling kept them from immediately adding to their tally.
And they had to make do with another Lawlor penalty from close in to stretch their advantage.
With victory apparently in the bag, Munster began to open out even more and it was no surprise when Lions centre Rob Henderson waltzed over for touchdown number three, goaled by Lawlor.
Glasgow replied in positive fashion with a Graeme Morrison try - and when Rory Kerr darted over five minutes later and Moffat converting, the game was back in the melting pot.
But Munster clicked into gear again to set up scores for Lawlor and Barry Murphy, both of which were goaled by the former.
Final Score: Glasgow 12-37 Munster
Glasgow: (12) Tries: Kerr, Morrison Conv: S Moffat
Munster: (37) Tries: Kelly, Lawlor, Henderson, McPhail, B. Murphy Cons: Lawlor 3 Pens: Lawlor 2
Metcalfe, Kerr, Morrison, A. Bulloch, Lamont, Howarth, Pinder, A. Kelly, Gunn, Murray, Beardshaw, Ross, Wilson, MacFadyen, R. McKay.
Replacements: Lawson, Proudfoot, Hall, Sitch, Beveridge, MacLure, A. Moffat.
Payne, Kelly, Lawlor, Henderson, McPhail, Holland, Reddan, Roche, Flannery, McIlwham, Hogan, Pusey, Halvey, McMahon, Williams.
Replacements: B. Murphy, O'Sullivan, F. Murphy, P. Malone, S. Keogh, Callaghan, Cahill.