The Scarlets dominated the first period but trailed 13-8 at the break but the second half was a complete role reversal as the Irish giants had the better of territory and possession but couldn't find a breakthrough, with Barclay's solo score and a stern defensive effort ensuring a repeat of last season's semi-final showing remains a possibility.
A third successive league win means they are now 10 points behind the fourth-placed Ospreys with a game in hand and seven matches still to play.
New signing Olly Barkley kicked the Scarlets in front with a close-range 13th-minute penalty but Munster were level when Ian Keatley slotted an equally straightforward penalty after good work from Ireland and Lions wing Simon Zebo, who started at full back for the visitors.
The Scarlets went close after a quarter of an hour but Dave Kilcoyne's huge hit on opposite number Phil John slowed the momentum after Johan Snyman had been dragged down five metres out. Munster could have capitalised in spectacular fashion seconds later when the ball popped loose at the resulting ruck but Kilcoyne's pass went behind Gerhard van den Heever, who would have backed himself to race 80 metres for a breakaway try.
Keatley's kick was the only other time the Scarlets were really troubled for much of the first half but the scores remained all square until Davies crossed for a superb individual try with 26 minutes played. The Scarlets scrum-half went blind from the back of a driving maul and rounded Munster captain and No8 James Coughlan before stepping off his right foot to leave Duncan Williams and Johne Murphy off balance, with Coughlan unable to get back in time to prevent the touchdown.
Barkley was unable to add the extras from five metres in from the right touchline but the Scarlets at last had some reward for a hugely encouraging performance against a side who had seen off the Ospreys last Sunday and hadn't lost in the league since mid-September.
But the half ended on a sour note as they were hit by a sucker punch from van den Heever before Keatley struck a second penalty with the final play. Van den Heever pounced on the loose ball when Barkley spilled Aled Thomas' pass on his own 10-metre line before racing home unopposed.
Keatley completed the seven-point score from underneath the sticks and the Scarlets were suddenly five points adrift when he rubbed salt into their wounds after John was penalised for going off his feet at a ruck on the edge of the 22 deep into added time.
Rob McCusker and co would have been 16-8 behind had Keatley landed another penalty just a minute after the restart but the scoreboard belonged to the Scarlets throughout the second half, even though it was Munster who were on top for much of the final 40 minutes.
Thomas cut the gap with his first penalty with half an hour remaining and the game suddenly swung the Scarlets' way when Barclay picked off Duncan Williams' pass before showing outstanding pace for a back rower to sprint clear from fully 60 metres. Thomas sent over the conversion to signal a 10-point swing and Munster were unable to find a way back in the final 28 minutes.
The visitors came close on a number of occasions, though, with van den Heever missing a brace of chances and seeing a score ruled out for a knock on from Williams in the build up. Sean Dougall also came close to levelling affairs when he was held up immediately after Scarlets blindside Josh Turnbull saw yellow for careless use of the boot on Duncan Casey with 12 minutes left but the Scarlets' defensive desire and unrelenting work ethic ensured they enter the two-week break on an almighty high.
There was good news for Wales head coach Warren Gatland as well ahead of the RBS 6 Nations trip to Twickenham a week on Sunday after centre Jonathan Davies played over an hour on his comeback from a pectoral injury.