The Scarlets had to hold on at the end but deservedly emerged with an 18-13 win away from home against a dogged Glasgow outfit.
Ruaridh Jackson slotted over a comfortable penalty to edge the home side in front early on but it was the Scarlets who produced the first real moment of quality to register the first try at Scotstoun Stadium.
Some fantastic offloading in close quarters by Matthew Rees and Andy Fenby created the opportunity for George North to stretch his legs down the left-hand touchline and he found Aaron Shingler on his inside to score in the corner.
Shingler went from hero to villain moments later as he was sinbinned for a high tackle but Glasgow were unable to capitalise on their one-man advantage and, once they were restored to their full complement, the visitors extended their lead before the break.
Jonathan Davies hacked a loose ball downfield after a handling error in the Glasgow backline and Fenby latched onto it to touch down in the left-hand corner.
Rhys Priestland failed with the conversion though and Jackson kicked a late three points to close the gap to just 10-6 at the interval.
The Scarlets were the better side for the most part though and a Priestland penalty extended the lead early in the second half before a good miss pass from Scott Williams enabled North to cross in the left-hand corner.
Chris Cusiter did give Glasgow some hope with a try under the posts with just a few minutes remaining after a great break and offload from Alex Dunbar but they had to settle for a losing bonus point as the Scarlets held on to record their second consecutive win.
Pens: Jackson 2
Tries: Fenby, North, Shingler
Glasgow: 15 Peter Murchie, 14 DTH van der Merwe, 13 Alex Dunbar, 12 Graeme Morrison, 11 Sean Lamont, 10 Ruaridh Jackson, 9 Chris Cusiter, 8 Ryan Wilson, 7 John Barclay, 6 Rob Harley, 5 Al Kellock, 4 Tom Ryder, 3 Moray Low, 1 Ryan Grant, 2 Pat MacArthur.
Replacements: 16 Dougie Hall, 17 Mike Cusack, 18 Gordon Reid, 19 James Eddie, 20 Chris Fusaro, 21 Henry Pyrgos, 22 Duncan Weir, 23 Peter Horne.
Scarlets: 15 Liam Williams, 14 George North, 13 Jonathan Davies, 12 Scott Williams, 11 Andy Fenby 10 Rhys Priestland, 9 Tavis Knoyle, 8 Rob McCusker (c), 7 Johnathan Edwards, 6 Aaron Shingler, 5 Richard Kelly, 4 George Earle, 3 Deacon Manu, 2 Matthew Rees, 1 Rhodri Jones.
Replacements: 16 Ken Owens, 17 Phil John, 18 Peter Edwards, 19 Sione Timani, 20 Kieran Murphy, 21 Gareth Davies, 22 Aled Thomas, 23 Adam Warren.
Referee: David Wilkinson (Ireland)