Scarlets had to do something in the second half and it took two minutes for Priestland to claw back three points.
They did try to inject some pace in the game, but found themselves up against a well-organised Munster defence and victims of their own unforced errors.
Priestland did pull them back to within a converted try of taking the lead with a penalty after Leamy was caught offside.
And Scarlets enjoyed more of the possession without really punishing their visitors.
But the game suddenly burst into life in the last seven minutes.
First Munster were again reduced to 14 as Donncha O'Callaghan was yellow-carded for preventing a quick penalty.
And then Scarlets thought they had stolen the victory with five minutes remaining, winning a penalty try after Munster's front row popped up in the scrum.
But Deacon Manu gave a needless penalty from the restart for a shove in the face and O'Gara sealed the win for Munster.
Ceiron Thomas had the chance to change that in the closing stages, but his drop-goal attempt was charged down and the Scarlets had to settle for a losing bonus point.
A further blow for the Scarlets was the loss of prop Kees Meeuws to an Achilles injury.
Scarlets coach Nigel Davies: "We gave them a good start and fought back well in the second half and got our just rewards.
"Unfortunately there was a key moment from the kick-off which we got wrong and lost the game.
"I'm fed up of coming second best and losing gloriously.
"I'm just hugely frustrated. The game was there for us, we won the game and lost it again."
Munster match-winner Ronan O'Gara: "Llanelli have a fighting spirit and came back hard on us and we were lucky to get another penalty.
"We are happy enough to get out of here with four points.
"It was rusty, but we won and that's the main thing."
Scarlets: Stoddart, Daniel, King, G. Evans, Davies, Priestlend, Martens, I. Thomas, Owen, Meeuws, Cooper, Day, N. Thomas, Easterby, Lyons. Replacements: C. Thomas for G. Evans (73), Cattle for Martens (58), John for I. Thomas (73), Schwalger for Owen (58), Manu for Meeuws (40), Edwards for N. Thomas (73). Not Used: McCusker.
Munster: Warwick, Howlett, Earls, Mafi, Dowling, O'Gara, O'Leary, Horan, Sheahan, Buckley, O'Callaghan, O'Driscoll, Ryan, Ronan, Leamy. Replacements: Pucciariello for Horan (65), Fogarty for Sheahan (60), Coughlan for Leamy (73). Not Used: O'Sullivan, Stringer, Murphy, Horgan.
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