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A lacklustre opening half was finally brought to life on the half hour by Borders player coach Gregor Townsend.
The Scottish veteran, who returned to Borders in the off-season, grabbed the first try of the game after cutting through the Munster defence before dotting down under the posts.
Charlie Hore added the conversion but Munster soon responded with a penalty from Paul Burke and wen tclose on two occasions to adding a try to their tally but were kept out by some good defence.
Burke added a second penalty after the break to narrow the gap and Hore missed an opportunity to restore their advantage.
After another miss, this time from Burke on the hour, it was the boot of replacement teenage flyhalf Jeremy Manning, a former New Zealand schools fly-half, that kicked what proved to be the match-winning penalty.
The narrow defeat means Borders are still without an away win in four seasons of Celtic League rugby.