Adrian Jarvis kicked his side in front after just two minutes, a score that was cancelled by Ian Keatley four minutes later with a penalty from near half way.
However it was Jarvis who won the first half kicking duel, he landed two more penalties while Keatley was wide of each side of the uprights with two efforts within two minutes.
A fine Luke O'Dea break early in the second half ended with a body check and a resultant yellow card for David McIlwaine, a penalty for Munster and a try for Dave Kilcoyne.
Ten minutes later Duncan Williams was over after Casey Laulala had punched a hole close in and finally Sean Scanlon scored his second try in as many games with five minutes of normal time remaining.
Munster: D Hurley; D Howlett capt (S Scanlon 61 min), C Laulala, J Downey (D Barnes 51 min), L O'Dea (J Murphy 61); I Keatley (S Deasy 61 min), D Williams (P Stringer 55 min); D Kilcoyne (W du Preez 51 min) (M Horan 70 min), D Varley (S Henry 47 min) (D casey 71 min), S Archer (BJ Botha ht); B Holland (P Donnellan 66 min), D Foley (B Hayes 47min); T O'Donnell, S Dougall (B O'Mahony 61 min), P Butler (Dave O'Callaghan ht).
Bristol Rugby: Jon Goodridge; Ryan Edwards, Bryan Rennie, Luke Eves, George Watkins (David McIlwaine); Adrian Jarvis, James Grindal; Kyle Traynor, Ross Johnston, Wayne Thompson; Mariano Sambucetti, Glen Townson; Iain Grieve capt, Redford Pennycook, Mitch Eadie. Replacements: Will Davis, Rhys Lawrence, Bruce Douglas, Roy Winter, Tom Slater, Jack Gadd, Marco Mama, Tristan Roberts, James Merriman.