Now on 62 points in fourth, Scarlets are 15 points ahead of Munster who can only amass the same amount from their final three games.
Tries from hooker Ken Owens and Liam Williams, as well as 19 points from the boot of outside-half, Owen Williams saw them home.
Scorers for Scarlets: Tries: Owens, L. Williams. Cons: O. Williams 2. Pens: O. Williams 4. Drop Goals: O. Williams.
Scorers for Glasgow: Pens: Jackson 2.
SCARLETS: L. Williams, North, J. Davies, S. Williams, Fenby, O. Williams, A. Davies, John, Owens, Lee, Earle, Snyman, Shingler, McCusker, J. Edwards.
Replacements: Maule for J. Davies (73), G. Davies for A. Davies (64), R. Jones for John (64), Rees for Owens (68), Manu for Lee (56), Timani for Earle (70), Turnbull for J. Edwards (58).
GLASGOW: Hogg, Maitland, Bennett, Horne, van der Merwe, Jackson, Pyrgos, Reid, MacArthur, Low, Ryder, Kellock, Strauss, Barclay, Wilson.
Replacements: S. Lamont for Bennett (49), Weir for Jackson (52), Matawalu for Pyrgos (56), Grant for Reid (32), Hall for MacArthur (58), Kalman for Low (58), Swinson for Ryder (49), Harley for Barclay (66).
Referee: George Clancy.