In excellent conditions, Owen Williams opened the scoring for the visitors within a couple of minutes with Dan Newton cancelling out that score 10 minutes later.
Play was evenly contested but Llanelli were looking the more threatening with ball in hand and when flyer Kyle Evans scorched in for another good individual try, converted by Owen Williams, things looked promising for the visitors.
Williams then put over another fine penalty for a 13-3 lead which remained until injury time of the half when Newton narrowed Quins' deficit.
Quins came out of the blocks well in the second half and Llanelli's fine defence finally cracked when Aled Hopkins was yellow carded by referee Jon Mason. Newton kicked the resulting penalty and soon after wing, Jason Harries crossed the visitors' line after good approach from his forwards. Newton added the simple extras.
Newton and Williams swapped penalties before a Johnny Lewis charge down resulted in a 50 metre dash and for the centre to give his team a 21-19 lead.
Quins replaced Newton with young Academy Scarlet player Jack Maynard, who showed great maturity to slot over what was to be the winning 35 metre penalty, on 70 minutes.
In the final 10 minutes Llanelli blew a number of chances to score tries and in injury time Williams' long distance penalty attempt drifted wide for the Quins to hold onto the win and gain the four valuable Principality Premiership points.