The victory however was marred by the sending off of prop Euan Murray who was given his marching orders for illegal use of the knee.
Glasgow stole the initiative with two early penalties from Calvin Howarth but the Scarlets responded with a well-worked try from Barry Davies that was converted by Gareth Bowen.
But from this point on Glasgow asserted themselves confidently to put the game out of reach for the Welsh side.
Scrum-half Graeme Beveridge grabbed the first five pointer after som egood work from Gareth Maclure and Scott Lawson, with Howarth slotting the conversion and adding a quick penalty.
Bowen clawed back three points, but the Scarlets' challenge wilted in the face of some stiff Glasgow pressure.
Logan stamped his mark on the game when he got on the end of a long Howarth pass to score under the crossbar. Then Howarth jinked over for a brilliant solo effort.
Logan soon grabbed his second, doing superbly to burrow his way over in the corner. Once again Howarth added the conversion.
He soon added another penalty before the Scarlets showed some sign of life with a long range try from replacement Salesi Finau, with Bowen adding the extras.
Scrum-half Mike Phillips followed suit as Glasgow took their foot off the pedal - and the jitters set in when Murray was dismissed.
They were reduced further as Fergus Thomson was sin-binned - and Garan Evans slid over to salvage a late consolation point for Llanelli.
|Glasgow Rugby Score Card|
|Llanelli Scarlets Score Card|