Victory for the Warriors would have ensured them a play-off spot but they now find themselves knocked off the top of the table by Ulster and far from guaranteed a home semi-final.
The Scottish outfit were never really at the races at Parc y Scarlets and inspired by the nerveless kicking off Owen Williams the Scarlets moved into fourth with Ospreys playing on Saturday.
In the third minute a clever backs move saw Liam Williams release Jonathan Davies down the right and the scrambling Glasgow defence was forced to concede a penalty.
Fly-half Owen Williams, who has appeared in the last six games for the Welsh region, boomed over the long-range kick and the Scarlets were 3-0 up.
The Warriors were really struggling to settle in the opening exchanges, Ruaridh Jackson's kick charged down while Peter Horne's pass went straight into touch.
But despite the dominant territory the Glasgow defence, so impressive all season long, held firm in the face of some considerable pressure.
But in the 13th minute it finally told and when Gordon Reid used his hands in the ruck Owen Williams, who is heading for Leicester Tigers in the summer, slotted his second penalty of the night.
Jackson was not enjoying the finest of starts and when his sliced clearance off his left foot went out on the full from the re-start Scarlets were again on the front foot.
Owens made the initial break into the 22 and after a series of pick and go drives around the fringes it was Owens again who burrowed under the posts for the first try of the game, Owen Williams adding the extras for 13-0 after 18 minutes.
The Warriors badly needed a foothold in the game and Jackson provided it on 22 minutes, slotting over a very tough penalty from near halfway to make it 13-3.
Owen Williams then slotted a drop goal from close range before a second Jackson penalty made it 16-6 on the half-hour mark.
John Barclay was one of the few Glasgow players to emerge with any credit from an under-par first half and when he won a turnover in front of the posts it was strange to see Jackson, who had made his last 16 kicks in a row for Glasgow, instead opt for a kick to the corner.
And that decision was made to look a poor one when Al Kellock knocked on in sight of the try-line and scrum-half Aled Davies kicked clear to relieve the pressure, leaving it at 16-6 at the break.
Three minutes after the interval and Owen Williams erred from the kicking tee for the first time, his penalty attempt going wide to let-off Gregor Townsend's side.
But their stay of execution did not last long as Liam Williams ghosted over for the Scarlets' second try moments later, collecting George North's offload for an easy score, Owen Williams' conversion making it 23-6.
A 51st minute penalty from Owen Williams increased that lead to 26-6 and when Duncan Weir, on as a replacement for Jackson, smashed a penalty against the post the Scottish side's challenge was all but over.
Now the only debate was whether the Welsh region could get a try bonus point but on the hour mark they strangely opted for the posts again rather than the corner and Owen Williams continued to impress, slotting another penalty for 29-6.
The final quarter of an hour was a somewhat flat affair as both sides appeared to have settled for the result before Andy Fenby made a clean break with three minutes remaining but his offload went to ground and the chance was lost.
Gareth Davies came even closer in the dying moments of the game but a fantastic covering tackle from Horne denied him and it finished 29-6.
|Scarlets Score Card|
|Glasgow Warriors Score Card|