A penalty apiece from Ian Keatley and Aled Thomas were the only scores of a slow-burning first half in weather-beaten Cork.
Thomas kicked the Welshmen ahead in the 68th minute before Andy Fenby pounced on a Felix Jones mistake to score his sixth try of the season.
Thomas added an excellent conversion, but Munster answered back in injury time with replacement JJ Hanrahan's penalty giving them a losing bonus point.
It took a while for the game to ignite, with a testing wind and French referee Laurent Cardona's whistle having the most influence.
Munster full-back Jones tested the Scarlets rearguard with a dangerous kick chase, and the first sniff of a try saw hooker Mike Sherry hauled down a couple of metres out.
Scarlets, missing 15 players due to international call-ups, absorbed the early pressure and had the first shot at the posts, but Thomas' wind-affected effort swung away to the left of the target.
Into the second quarter, Keatley and Thomas swapped penalties, with both sides guilty of unforced errors and static play.
The changes in personnel were an obvious factor and a long injury stoppage for Niall Ronan late in the first half also broke up the momentum.
Ronan suffered a blow to the head as Jake Ball hurtled into a ruck near the halfway line, an incident which may result in the young Scarlets lock being cited.
Munster pressed for a late try and Ronan's replacement, South African CJ Stander, was held up on a close-range drive in injury time.
With the rain now teeming down into Munster's faces, the Scarlets resumed in better fettle and Ball's charge down almost led to the onrushing Jonathan Edwards scoring.
Thomas slid a penalty wide, failing to punish Munster for a scrum infringement. Spilled passes were a regular occurrence as conditions worsened.
A drop goal shot from Thomas went the wrong side of the left-hand post, and the Munster forwards responded with a strong spell of carrying and direct running.
Jones lifted the home crowd with two surges from deep, the second of which leading to Keatley hacking through and the Scarlets had to scramble back close to their line.
The Welsh side edged ahead with 12 minutes remaining, Thomas planting a penalty over with the aid of the left upright amid a strong period of scrummaging.
Led by man-of-the-match Emyr Phillips, Simon Easterby's men held the upper hand in the scrum and continued to win decisions from Cardona.
Munster handed them a very soft try shortly afterwards.
The Scarlets launched a kick chase after Luke O'Dea had coughed up possession in midfield, the retreating Jones getting two bites at the loose ball but seeing it squirt free over the whitewash for Fenby to flop onto it.
Thomas' successful conversion put the seal on the Scarlets' first victory over Munster in 16 meetings, a run stretching back to 2007.
The hosts earned a late bonus point, though, as Hanrahan split the posts with the final kick of the game.
Ceri Sweeney kicked nine second-half points to give Cardiff a 14-7 victory over Zebre which ends their miserable recent run.
The Welsh side had lost their last four games, but Sweeney's accurate boot enabled them to see off the competition newcomers, who are still seeking their first win.
The Italians led 7-0 lead through Samuele Pace's try midway through the first half, taking advantage of the extra man after Robin Copeland had been sent to the sin-bin.
Within three minutes of Copeland's 12th-minute departure, centre Pace ran through to score a try which was converted by Daniel Halangahu.
Cardiff responded in the 27th minute as prop Nathan Trevett bundled over, although Sweeney was not able to take advantage with the conversion.
The hosts went in with a 7-5 advantage, although the gap would have been five points had Gonzalo Garcia not sliced his penalty wide at the end of the first half.
Cardiff's Mike Paterson came on for Macauley Cook five minutes after the restart for his first appearance of the season, following operations on both shoulders.
The visitors responded well to the return of their player of the year, with Sweeney edging them into an 8-7 lead in the 56th minute.
Although the 32-year-old missed another attempt five minutes later, he kept his nerve with kicks in the 68th and 70th minutes completing the turnaround and lifting the Blues to seventh in the table.