Wing Tal Selley, centre Matthew J Watkins and prop Iestyn Thomas all crossed the try line for the visitors, while outside-half Gareth Bowen contributed with 13 points from the boot.
An unimaginative Tykes, who have failed to win a Guinness Premiership match this season, had failed to breach the Scarlets defence until substitute Rob Rawlinson scored a consolation try in the final quarter of the second half. Making his debut for the Tykes was 20-year-old England Sevens international Rob Vickerman. He replaced Andre Snyman, who had returned to South Africa for family reasons, at centre.
Captain Stuart Hooper, celebrating a call-up to the 30-man England squad ahead of the autumn internationals, and Gordon Ross also returned to bolster a side that had previously lost to Leicester Tigers. The Scarlets, who had also suffered defeat in their last match, welcomed back Lions scrum-half Dwayne Peel for his first competitive appearance of the season. Both sides looked intent on attack in the opening minutes and it was the Scarlets who took the early lead through a third-minute Bowen drop goal.
They doubled their lead with a Bowen penalty in the 21st minute and could have gone further ahead minutes later but for bad hands from Lee Byrne. A delicate Regan King pass allowed fellow centre Watkins to cut through the Leeds defence, but with the try line looming the full-back failed to take Watkins' inside pass.
The rest of the half, disjointed by the referee's whistle, was an uninspiring advert for the reconstituted Cup. Leeds, lacking confidence after a winless September, rarely threatened the opposition half with Tykes Academy graduate Vickerman the only back to show any intent on breaking the Welsh side's defence.
While the visitors, buoyed by their boisterous fans who had made the trip north despite the late kick-off, failed to make the most of their overwhelming possession. In stoppage time Watkins looked to have danced past the Leeds defence for the first try of the match but the referee blew for offside with the line approaching and so the score remained 6-0 at half-time.
The Tykes had the chance to gain their first points in the 53 minute after Peel was penalised but outside-half Gordon Ross failed with his kick. Leeds were to rue their profligacy minutes later when wing Selley scored for the Scarlets' first try after good work by Bowen and Watkins. Bowen missed the touch-line conversion but was successful on the hour with a penalty, which gave the visitors a comfortable 14-0 lead.
The Welsh region went further ahead in the 65th minute when powerful forward play and a darting Peel run allowed prop Thomas to cross for the try, converted by Bowen. Another Scarlets converted try minutes later, when Watkins intercepted full-back Iain Balshaw's pass to score under the post, gave the visitors a commanding 28-0 lead.
Leeds finally managed to get on the scoresheet when Rawlinson crossed underneath the posts, substitute Roland de Marigny, on for Ross, converted. Substitute number eight Gavin Quinnell was sin binned in the dying minutes but it didn't hamper the Scarlets win.