An outstanding charge through the middle by prop Nicky Smith earned the visitors a penalty when the Quins defence stepped offside, but Nicky J Thomas pulled his effort wide.
The home side got the first try when flanker James Davies showed great strength to step out of a tackle and beat the first man before passing to centre Andrew Banfield, whose pace did the rest. Maynard converted.
Nicky J Thomas got his side on the board with a penalty when the Carmarthen defence were caught offside in midfield. The full back had an opportunity to double his side's points just a minute later when the Quins were penalised for straying in front of a kick, but he missed his shot at goal.
The second Quins try was a joy to watch with the backs and forwards combining well. Firstly through the power of Luke Joseph, who broke through a tackle to set up the space, then centre Rheon James was on hand to stretch the Whites' defence, before he found Banfield who sprinted down the wing to score. Maynard converted.
Signs of ill-discipline began to show in the Quins ranks and Davies was sent to the sin-bin for persistent offside. It was what the visitors needed and scrum half Jake Cole took full advantage by weaving his way over for a try after a routine lineout take by the pack. Nicky J Thomas converted.
Five minutes later though Cole was shown the yellow card for killing a dangerous Carmarthen attack. This time it was the home side who took advantage of the sin-bin with Maynard managing to keep his balance to dive over the line after another well taken lineout. He also converted.
The home side got their bonus point inside first half stoppage time after a spectacular counter attack orchestrated by Joseph and Banfield resulted in a try for scrum half Connor Lloyd. Maynard converted to give the home side a 28-10 lead at the break.
In the second half the home side looked to continue as they had left off in the first when winger Jason Harries eased through three tackles out wide only for Lloyd to fumble at the ruck.
The Quins kept up the pressure inside the Swansea twenty-two and despite an initially solid defensive line by the Whites they stepped offside so Maynard was able to add another three points.
Powerful running by Whites replacement Scott Otten and captain Sam Kiley created a rare attacking platform and Nicky J Thomas kicked a penalty to the corner. Carmarthen flanker Aidan Laxton was then yellow carded for persistent offside.
Swansea got themselves back to within two scores when replacement forward Dan Baker charged down a poor attempted clearance by Huw Rees and gathered to cross the line. Nicky J Thomas converted.
The home side regained the upper hand and Rees made up for his mistake by earning his side a penalty after he made a good break through the middle. When the Whites failed to retreat 10 metres referee Cammy Rudkin showed replacement prop Matthew Norman a yellow card.
Nicky Harwood did well to steal possession near his own try line but the centre was then penalised for holding onto possession in the tackle. Maynard converted the penalty goal.
The Quins rounded off a fine performance with a sixth try after five minutes of solid Whites pressure ended with a mistake and James Davies capitalised by stealing possession to race in to score. Maynard converted.
The two sides meet again next week at St Helen's in a Principality Premiership match.