And despite an improved showing in the second half from the Warriors, who lost fly-half Ruaridh Jackson to injury inside the first 15 minutes, Scarlets held firm for their third win in a row that puts them fourth with Munster playing in the late kick-off.
Meanwhile the Warriors, who scored the first try of the game through Moray Low, were forced to settle for a losing bonus point that could yet prove vital.
After some cagey opening exchanges it was Sean Lineen's side who seized the initiative in the seventh minute when prop Low crashed over after Scarlets made a hash of a lineout near their own try-line.
Jackson failed to add the extras and Rhys Priestland reduced his side's arrears to only 5-3 when he slotted his first penalty of the match just three minutes later.
But in the twelfth minute the gap was back out to five when Jackson slotted over a penalty after the Welsh region failed to roll away at the breakdown.
But that was Jackson's final contribution as he limped off with an apparent leg injury after the kick to be replaced by youngster Scott Wight.
Two penalties from Priestland then turned the match around and the Scarlets led 9-8 after half an hour and it wasn't long before Glasgow's ill discipline really began to cost them.
In the 36th minute the Warriors found themselves reduced to 13 men as first Rob Harley and then Ryan Wilson both saw yellow for separate incidents, Harley involved in an off-the-ball altercation and Wilson pinged for not releasing.
But despite their numerical disadvantage it looked like the Warriors might hold on until half-time but the Scarlets had other ideas and on the stroke of the whistle Davies snuck over to make it 16-8.
The pressure continued after the interval as the home side looked to make their extra numbers count but some stout defending from Glasgow kept them at bay, and the Warriors actually reduced their arrears on 44 minutes when Wight slotted a penalty from near the halfway line to make it 16-11.
Wight then missed with a simpler effort just before the hour mark as the visitors, restored to 15 men, pressed in search of a way back into the game.
The Scarlets had their chances as well and Calum Forrester had to be at his very best to deny Jonathan Davies a try with a last-gasp tackle to keep the score at 16-11.
Wight then reduced the Scottish outfit's arrears further with a penalty with 15 minutes to go but try as they might the Warriors could not find that final breakthrough and Nigel Davies' men held out for what could prove to be a crucial victory.
|Scarlets Score Card|
|Glasgow Warriors Score Card|