The sides meet again at Goldington Road next week and Munster simply have to win to have any chance of topping their group and challenging for a home quarter final.
They could lose next week and still finish in the last eight by beating both Nottingham and Ospreys in the final set of games, but they will need to show significant improvement from this poor second-half performance to restore confidence.
Munster had a dream start when they scored in the opening minute, winger Jack Power grabbing the try after a deft chip ahead by James Hart and a mistake by a Bedford defender. Hart kicked the conversion from the touchline to establish a 7-0 lead.
The home side continued to force the pace in that opening spell and were unfortunate not to add at least another try even in the face of some excellent defending from the visitors, whose only scoring chance came in 21 minutes when full back Rich Lane lost the ball forward in a good attacking move.
But Munster did get their second after a series of attacks immediately afterwards and Alex McHenry was sent in under the posts after 13 phases for a try that Hart converted again.
Against the run of play, Bedford got back into the contest with a 33rd-minute try from loose-head prop Sean McCarthy and outhalf Will Hooley kicked the difficult conversion to make it 14-7.
The visitors were back on level terms two minutes after the break when danger man, left winger Elliott Clement-Hill, scythed his way through for a try at the posts that Hooley converted and the out half added 51st and 60th minute penalties to give his side a 20-14 lead.
Munster launched a series of furious assaults in a bid to save the game but paid for their handling mistakes, having trouble in the scrum and they lost one crucial line out on their own throw when trying to overhaul a six points deficit.
From there, Bedford scored another try, albeit against the run of play, from Clement-Hill and that was enough to deny Munster a losing bonus point.
S Fitzgerald; J Power, D Johnston (C), A McHenry, L Coombes; B Johnston, J Hart; J Conway, M Sherry, J Loughman; F Wycherley, S McCarthy, S O’Connor, T O’Donnell, G Coombes.
J O’Sullivan for S O’Connor (40), D Casey for Sherry, B O’Connor for Conway (both 54), J Poland for Hart., D McCarthy for McHenry (both 56). J Foley for G Coombes (74), C Fitzgerald for B Johnston (78).
R Lane; E Clement-Hill, O Dodge, M LeBourgeois (captain), D Adamson; W Hooley, J Burns; S McCarthy, T Lindsay, B Cooper; E Taylor, J Onojaife; A Rae, J Buggea, J Williams.
W Carrick-Smith for Rae (27, injured), H Worthington for Taylor (44, injured),J Fields, A Penny for Cooper (60), L Dickson for Burns (67). H Sheppard for Hooley, D Spelman for S MccCarthy (both 79)
Stade Francais 39 Krasny Yar 24
Agen (14) 21 Pau (21) 40
Newport Gwent D’gons 15 Enisei-STM 0