The boot of Gareth Bowen rescued the Scarlets at Stradey Park on Saturday.
The Scarlets went into the match desperate to banish memories of the last time Glasgow visited Stradey Park - a 57-30 drubbing in April - but produced only glimpses of the sort of form needed to compete in Europe.
In-form fly-half Bowen (three) and opposite number Dan Parks (two) traded penalty goals for the first 30 minutes, before Glasgow scrum-half Sam Pinder was sin-binned for a professional foul. The Scarlets took immediate advantage and huge Springbok lock Hottie Louw lumbered over the line for the game's first try on the stroke of halftime.
Parks narrowed the margin with his third goal, before young Scarlets scrum-half Liam Davies provided a much-needed spark to a game in danger of falling asleep. He took the ball at the base of a ruck five metres out, stood up the final defender with the confidence of a 50-cap veteran and darted over in the corner.
The fans' joy was short-lived as replacement Glasgow backrower Gregor Hayter ran 20 metres untouched for a try that Scarlets defence coach Paul Moriarty will have nightmares about, and Parks' fourth goal narrowed the margin to four points.
The two kickers traded penalties as the clock ticked down, but some desperate defence in the eight minutes of injury time helped the Scarlets finish the match ahead on the scoreboard.
Llanelli Scarlets: L Byrne (G Evans 67); D James, M Watkins, R King, M Jones; G Bowen, L Davies (C Stuart-Smith 67); I Thomas, A Gravelle (M Rees 40), R Davies, I Afeaki (C Wyatt 50), H Louw (A Jones 50), J Mills, A Popham, S Easterby (capt).
Scorers: Tries - H Louw, L Davies; Pens - G Bowen 4; Con - G Bowen.
Glasgow: R Lamont (C Gregor 85); M Roberts, A Craig (G Morrison 57), A Henderson, H O'Hare; D Parks (capt), S Pinder (G Hayter 46); L Harrison, S Lawson (F Thomson 72), E Murray (K Tkachuk 40), C Hamilton, T Barker, P Dearlove (S Swindall 64), A Wilson (G Beveridge 40), J Beattie.
Scorers: Try - G Hayter; Pens - D Parks 5.
Halftime: Scarlets 16-6
Yellow card: S Pinder (Glasgow)
Referee: Donal Courtney (Ireland)