Adam Warren, Kirby Myhill, Richie Pugh and Kieran Murphy all touched down for the Scarlets with fly-half Aled Thomas adding ten points with the boot.
The visitors surged into an early lead straight from the kick-off. The Blues failed to deal with the kick and the Scarlets piled on the early pressure which resulted in Warren crashing over after two minutes.
Thomas converted and then slotted a penalty to stretch his side’s lead before Myhill scored the Scarlets second try.
Rhys Patchell grabbed a penalty for the hosts before the break but the Scarlets went into half-time 15 v 3 ahead.
The Blues came out fighting in the second period but once again the scoresheet was all Scarlets.
Thomas kicked a second penalty before Pugh burrowed over just before the 70-minute mark.
Murphy, who scored a brace last weekend against Leicester then wrapped up the bonus point with his try five minutes from time.
