In a tightly contested and hard-fought battle, Munster had the bulk of the territory and possession but could not convert that dominance into points as they were met by an excellent defensive effort from Scarlets, who also won numerous turnovers.
In the end, the Welsh region got the rub of the green with left wing Ioan Nicholas crossing for the game’s only try and their other points came via the boot of Wales full-back Leigh Halfpenny, who slotted a conversion and a penalty.
For the visitors, Bill Johnston succeeded with two penalties.
Munster were fastest out of the blocks but despite doing most of the attacking during the game’s early stages, they had little to show for their efforts as they were kept at bay by Scarlets’ miserly defence.
The Irish province took the lead in the fifth minute, however, courtesy of a three-pointer off the kicking tee from Johnston after Scarlets’ front-row were blown up for illegal scrummaging.
Six minutes later, the home side struck back via Nicholas’ try after gathering a brilliant offload from Johnny McNicholl deep inside Munster’s 22.
Halfpenny added the extras before Johnston replied with his second penalty on the stroke of half-time, which meant Scarlets held a slender 7-6 lead at the interval.
The second half was a slugfest with both sides piling on the pressure but a 47th minute penalty from Halfpenny were the only points scored during that period.
The game’s closing stages were frantic with Munster giving their all in a bid to regain the lead but they could not add to their points tally as Scarlets kept them at bay with solid defence.
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For Glasgow, Matt Fagerson (2), Chris Fusaro, Stafford McDowall, Ruaridh Jackson and Tim Swinson scored tries, and Adam Hastings kicked six conversion.
Zebre’s points came via a try from Tommaso Castello while Francois Brummer added a conversion and Carlo Canna slotted a penalty.
Antonio Rizzi led the way for the Italian outfit with a 10-point haul courtesy of a try, a conversion and a penalty and their other points came via a try from Niccolo Cannone and a penalty from Ian McKinley.
For Edinburgh, James Johnstone scored a try and Jaco van der Walt added a penalty and a conversion.
Elsewhere, Connacht thrashed the Ospreys 46-5 at the Sportsground in Galway. Kieran Marmion (2), Thomas Farrell, Matt Healy and Tom McCartney crossed the whitewash for the Irish province, who were also awarded a penalty try. Kyle Godwin contributed 14 points after succeeding with four conversions and a penalty.
Saturday’s final PRO14 match, between Cardiff Blues and the Kings, saw the Welsh region run out 26-19 bonus-point winners thanks to scores from Owen Lane, Nick Williams, Olly Robinson and Jason Harries. Dries van Schalkwyk, Michael Willemse and Masixole Banda responded for the Kings, who leave Cardiff with a bonus-point.