Five successes out of six penalties in torrential conditions at Parc y Scarlets were enough to hoist the home side into a seemingly impregnable 15-0 lead early in the second half, but then the picture changed within minutes.
Lock Lou Reed was given a yellow card for not rolling away quickly enough at the tackle area and it took Glasgow, who had beaten the Ospreys five days earlier, just three minutes to take advantage of their extra man.
Having built up a head of steam as the rain eased at the start of the second half, the visitors rung the changes in their side and replacement outside half Duncan Weir made an instant impact.
The teenager weaved his way past four defenders in a great run from his own 10 metre line that ended at the posts with a try that he also converted. That gave Glasgow the impetus they needed and four minutes later they made Reed feel even more uncomfortable on the sidelines.
This time it was Scottish international centre who made the initial thrust and his pass to left wing DTH van der Merwe allowed the Canadian star to run to the posts for a second try which Weir again improved.
Now there was only one point between the two teams with 23 minutes left on the clock. Either side could have won the game at that point, but Reed's return helped to steady the ship and a sixth penalty from Priestland gave the Scarlets a bit of breathing space.
"I was starting to panic 60 minutes into the game and I thought we had hit the self destruct button," admitted Scarlets skipper for the night Iestyn Thomas.
But there was enough composure in the end for the Scarlets forwards to hang on to win the game and confirm their status as the most improved team in the Magners League. This was their fifth win from seven starts, matching the number of victories they managed throughout the whole of last season. Then they mustered 29 points from 18 games and now they have 24 from seven.
"It was a very tough game, but that's a very good Glasgow side and I am delighted with the result. We haven't beaten them for two seasons," said Scarlets coach Nigel Davies.
"I was absolutely delighted. The back row were fabulous, Damian Welch put up his hand and Rhys Priestland is playing fabulous rugby.
"They may not be our big names, but they are our stars this season and our stars of the future."