The Scots looked on course for a second Celtic League win six games and a first away success of the season, after running in three tries - including two from openside flanker Daniel Kane - on their way to a 21-8 half-time lead.
But Munster, with eight players away preparing for Six Nations duty, hit the front by the hour mark - outside-half and captain Jason Holland pulling the strings.
A third-minute kick from scrum-half Eoin Reddan saw Borders concede an early penalty for full-back Jeremy Staunton from 45 metres.
Staunton met opposite number Scott Paterson in a try-saving tackle 10 minutes later, before Shannon hooker James Blaney touched down by the right touchline.
A lapse in the home side's concentration saw Kane lurch over for his first score six minutes later - with centre Gareth Morton converting and then adding the extras again to second row Scott MacLeod's effort following a timely offload by fly-half Tanner Vili.
A shock win looked on the cards when Kane got in under the Munster posts in the 32nd minute, Morton hitting his third success for the 13-point lead.
But Munster clicked into gear in the second half, the addition of backs Mike Mullins and Dominic Crotty seeing them attack Borders at will.
Horgan ran effortlessly under the posts 10 minutes after the break - with Holland converting - and he then slashed open the Borders rearguard on the hour mark to score a great individual try.
Holland's conversion gave Alan Gaffney's men their one-point lead with 18 minutes left.
Munster survived a late scare when Vili's attempted drop goal was charged down by Holland, and centre Rob Henderson crossed in the seventh minute of stoppage for a converted try and an unlikely bonus-point win.
15 - Jeremy Staunton, 14 - Mossy Lawler, 13 - Jason Jones-Hughes, 12 - Rob Henderson, 11 - Anthony Horgan, 10 - Jason Holland (Capt), 9 - Eoin Reddan, 1 - Simon Kerr, 2 - James Blaney, 3 - Gordon McIlwham, 4 - Trevor Hogan, 5 - David Pusey, 6 - Peter Malone, 7 - David Wallace, 8 - Stephen Keogh.
Replacements: Frank Roche, Jerry Flannery, Donal Sheehan, Tom Hayes, Frank Murphy, Mike Mullins, Dominic Crotty.