They ran in five tries with Jack Maynard adding two conversions and a penalty as the west Wales outfit moved to within one step of the Millennium Stadium final.
Maynard gave the visiting Quins an early 3-0 lead with just seconds on the clock before Richard Wilkes got the scoreboard moving with his side's first try of the afternoon.
Quins' fly half Maynard added the conversion and after making a fast start, Carmarthen led 10-0 in as many minutes.
And with all the momentum, Sion Bennett and James Dixon added further first half scores as the visitors moved out of sight. Maynard converted the first of those two scores and while Blackwood did manage a score of their own, Quins led 22-7 at the break.
Lock Marc Williams crossed for Blackwood's try with Paul Emanuelli adding the conversion.
With a significant half time lead, the second half was quieter but Quins still managed to add two further scores.
Rheon James added his team's fourth try in the 47th minute and Haydn Pugh completed the scoring with seven minutes remaining.
Wing Rhodri Usher responded for Blackwood with Emanulli again successful with the conversion but it was not enough to send them through to the next round.