Stephen Jones starred for the Scarlets as they came away from Hughenden with a win to move to second spot in the Celtic League.
Glasgow (13) 18
Tries: S Lamont, J Beardshaw. Con: C Howarth. Pens: Howarth (2).
Scarlets (18) 31
Tries: J Davies, T Selley, L Byrne. Cons: S Jones (2). Pens: S Jones (4).
The Wales fly-half kicked 16 vital points and proved a slick link between the dominant visitors' pack and their talented back-line.
The Scarlets tries came from prop John Davies and centre duo Tal Selley and Lee Byrne.
Glasgow remain rooted to the bottom of the league, and tries from wing Sean Lamont and substitute Joe Beardshaw will be little consolation as they prepare for the Heineken Cup campaign.
The Welshmen stormed
into the attack from the outset and threatened to blow Glasgow away in quickfire
fashion, but a combination of desperate home defence and crucial spilled passes kept
the damage to a minimum.
Jones was inches wide with a long-range penalty after Andy Henderson
wandered offside, but made instant amends by setting up the first try for
Selley, who had plenty of time and space to cruise in from 35m out.
Glasgow had another scare when Glenn Metcalfe sent a loose pass into his own
danger-zone, but Rory Kerr spared his blushes by scooping it up and clearing.
The home side eventually strung together a series of raids, but could only muster a
penalty by Calvin Howarth for their efforts.
Selley profited from Stephen Jones' creativity
The Scarlets replied with a Jones penalty before tighthead prop
John Davies was driven over for their second touchdown, Jones slotting the
conversion from a tricky angle.
Glasgow hit back with a brilliantly worked try by Lamont.
Graeme Beveridge wrong-footed the Llanelli markers with a dash to the narrow
side before Lamont ducked inside two challengers to dive across.
Howarth slotted the extras to narrow the gap to just five points, the margin remaining as Jones and Howarth exchanged
penalties before the break.
Two more Jones penalties in the second half seemed to take the game out of Glasgow's reach, and when Byrne got the visitors' final
try the result was sealed.
Beardshaw ploughed over for a Glasgow try towards the end, but it was too
little, too late.
Scarlets team manager Anthony Buchanan was delighted by the form shown by his side as they prepare for the Heineken Cup next week.
"We started off with real intent, laid a forward platform, played it wide and took our opportunities - the backs looked really sharp," he told BBC Sport Wales.
"It was evenly balanced at half-time, but we took control territorially in the second period to secure a good win.
"The manner in which we played was very important and everyone involved is delighted.
"It's all about confidence and winning and with that result we look forward to welcoming Northampton on Friday night."
Glasgow: Glenn Metcalfe; Sean Lamont, Graeme Morrison, Andrew Henderson, Rory Kerr; Calvin Howarth, Graeme Beveridge; Euan Murray, Gordon Bulloch, Lee Harrison, Andy Hall, Nathan Ross, Cameron Mather (capt), Donnie Macfadyen, Jon Petrie.
Relplacements: Lawson, A Kelly, Beardshaw, Reid, Pinder, Parks, MacLure.
Scarlets: Barry Davies; Garan Evans, Tal Selley, Matthew Watkins, Salesi Finau; Stephen Jones, Dwayne Peel; Iestyn Thomas, Robin McBryde, John Davies, Vernon Cooper, Chris Wyatt, Dafydd Jones, Simon Easterby, Scott Quinnell.
Replacements: Phil John, Aled Gravelle, Adam Jones, Ian Boobyer, Mike Phillips, Gareth Bowen, Lee Byrne.
Referee: Alan Lewis (Ireland)