Hooker Huw Dowden crossed for the decisive score in the second half as the conditions limited the amount of running rugby on show.
It turned into a battle of the forwards with little between the two packs but Dowden went over in the left hand corner to put Pontypridd into a seven point lead.
Quins refused to lie down though and got back into things with a late, late score from full back James Dixon.
It left Carmarthen needing the conversion to send the tie into extra time but replacement Jack Maynard put the difficult kick wide as Pontypridd progressed to the Millennium Stadium final on May 4.
Earlier, man of the match Dan Godfrey and Alex Pickersgill had exchanged first half scores, the Pontypridd No. 8 opening the scoring with just seconds on the clock. Pickersgill got Quins back into it with a charge down effort and with Dai Flanagan and Dan Newton booting the rest of the points, Pontypridd led 13-11 at the break.
But in a scrappy second half where the boot was dominant, Dowden's try sent Quins to a narrow defeat.
The game had started with a flurry of action and it was just a matter of seconds before Pontypridd opened the scoring. With Quins prepared to give the Premiership leaders plenty of space, Godfrey needed no second invitation as he powered through for the first try of the day.
Flanagan converted from wide out and after seeing a long range penalty effort then clip the right hand upright, he soon extended his side's lead with a snap drop goal.
With possession inside the Carmarthen half, a poor pass from scrum half Gareth James looked to have stalled Pontypridd's momentum but Flanagan collected superbly and drilled over a low drop goal for a 10-0 lead.
If Pontypridd's fast start had shocked the Quins, they soon found themselves back in the game when Pickersgill dotted down. Pontypridd full back Adam Thomas saw his attempted clearance charged down and the Quins centre picked up the loose ball to score with ease. Newton misjudged the wind as his conversion attempt went wide but his side were now up and running.
After two early tries, the game then settled down as the wet and windy weather began to play a part. Quins showed their intent with a couple of powerful scrum surges but both sides began to make mistakes, perhaps understandably given the conditions.
Despite a number of passes falling to ground, it was Quins who were the better at stringing their attacking game together and when Pontypridd were penalised at the breakdown, Newton further reduced the deficit with a simple penalty.
With a place at the Millennium Stadium at stake, it was tense stuff and there was little between the sides at the break, Flanagan and Newton exchanging further penalties as Pontypridd led by just two points.
The second half started as the first had ended with plenty of spilt balls and both sides kicking for field position. It was Pontypridd who were having the better of the territory though and despite conceding the occasional penalty, they were retaining the ball impressively in worsening conditions.
Both sides made a number of changes, Joel Raikes replacing James at scrum half for Pontypridd, but after 20 minutes of the second half, it remained 13-11.
With the game dominated in the middle third, it seemed as if each side was waiting for the other to make a mistake. Ed Siggery nearly made the breakthrough, his blindside break catching the Quins off guard.
The flanker was tackled into touch but from the resulting line-out, Pontypridd drove forward and Dowden crashed over for what looked like the decisive score. Flanagan missed the difficult touch line conversion but with a seven point lead, it was now a tough ask for the Quins to come back.
That they did though and after pounding the Pontypridd line, Dixon crept over with just seconds remaining. With Newton injured in the build-up to the try, the pressure fell on Maynard to convert and send the tie to extra time with his first and only touch of the game.
Sadly for him, the kick went wide as Pontypridd stayed on course for a league and cup double.