The Black and Ambers had the advantage of a first-half wind at their backs, but a long-range try to Quins centre Emyr Jones allowed his side to turn around only 11-14 down at halftime.
Quins fly-half Giles Thomas added his third and fourth penalty goals, one of which was cancelled out by opposite number Griffiths, and the sides entered the final 10 minutes locked at 17-all.
Griffiths kept his nerve to slot the winning goal in the 73rd minute - as he did in the first-round match between the two sides.
Neath kept in touch with Newport with a predictable 35-12 victory at bottom-placed Ebbw Vale, with their cause being helped by two penalty tries.
Promising fly-half Matthew Jones added 15 points with the boot as the Welsh All Blacks kept alive their hopes of a League title.
At the World of Hurt ground in Aberavon, the third-placed Wizards trounced a woefully out-of-form Pontypool 26-8, with the Division's leading scorer Jamie Davies scoring 11 points to take his season tally to 226.
Caerphilly came back from 7-13 down at halftime to pip Swansea 17-16 at Virginia Park and overtake them in the table.
Replacement flanker Siope Polutele scoring the crucial try under the posts to give centre Stuart Thomas a simple conversion to win the match.
Llanelli's hopes of climbing out of the bottom four weren't helped by an 18-30 loss to Bedwas at Stradey Park, while Pontypridd's solid season continued with a 27-21 victory at Llandovery.
Cardiff continued a great weekend for the Arms Park club, following the Blues' 50-10 Celtic League win over Edinburgh with a 25-5 victory at home to Cross Keys. Winger Elgan Jones scored two tries as Cardiff recovered from being held 5-all at halftime to record a convincing win.
Results: Aberavon 26 Pontypool 8, Bridgend v Newbridge postponed, Caerphilly 17 Swansea 16, Cardiff 25 Cross Keys 5, Carmarthen Quins 17 Newport 20, Ebbw Vale 12 Neath 35, Llandovery 21 Pontypridd 27, Llanelli 18 Bedwas 30.