The Scarlets survived a furious second half fightback from Glasgow at Stradey.
Scarlets (16) 24
Tries: Louw, L Davies
Pens: Bowen (4)
Glasgow (6) 20
Pens: Parks (5)
Tries from Hottie Louw and young scrum-half Liam Davies, plus effective kicking from Gareth Bowen, put the home side comfortably in control at 21-9.
But Gregor Hayter was let in for a soft 51st-minute try and from then on it was all Glasgow, Dan Parks' 15 points putting them within a score of victory.
The Scarlets held on, Glasgow taking a bonus point but seeing Andrew Wilson carried off with a dislocated ankle.
The early exchanges seemed a softening-up process, with the Scarlets dominating territory and possession and Bowen rattling up the points from three resultant penalties.
Glasgow fly-half Parks replied with two penalties of his own, though, to keep his side in touch.
But under the continual pressure the visitors' scrum-half Sam Pinder was yellow carded for tripping Davies at a ruck.
The Scarlets kicked to the corner, and from a scrum near the Warriors' line Springbok lock Louw ploughed over for his first try for the Llanelli side.
That made it 16-6, and to compound Glasgow's woes flanker Wilson was stretchered off on the stroke of half-time.
Parks landed another penalty early in the second half, but after a sustained Scarlets attack Davies - making his first start for the Scarlets - broke swiftly down the blind-side and had the speed to cross for the second try.
Glasgow replacement Hayter then hit the Scarlets with a shock score, taking short ball from a line-out on their 22 and brushing aside Matthew Rees' weak tackle for an unopposed run-in.
Hottie Louw got his first try for the Scarlets
Another Parks penalty cut the gap to just three points, as the Scots staged an inspiring fightback.
Garan Evans made his first appearance of the season in place of Lee Byrne, whose shoulder injury could make him doubtful for the Toulouse game.
Bowen and Parks exchanged penalties, and the Scarlets held off the Warriors' last desperate attacks.
To compound Glasgow's woes, Rory Lamont limped off with a leg injury in the last minute and could be doubtful for Friday's game in Bourgoin.
Scarlets scrum-half Dwayne Peel says he has an outside chance of making next week's game with Toulouse after his successful ankle operation.
He will definitely be fit for the following week's game with Edinburgh, when flanker Dafydd Jones also expects to be back.
Man-of-the-match Liam Davies:
"I really enjoyed my debut, it was great of the coaches to give me a chance.
"Crossing the line like that on my first appearance was something to remember."
Glasgow coach Hugh Campbell:
"It's very frustrating as it's another game we could have won.
"We just didn't get into it in the first half and gave away far too many penalties, especially at the rucks, and we just allowed them to dictate the territory and possession."
Scarlets: Byrne, James, Watkins, King, M. Jones, Bowen, L. Davies, I. Thomas, Gravelle, R. Davies, Afeaki, Louw, Mills, Easterby, A. Popham.
Replacements: G. Evans for Byrne (67), Stuart-Smith for L. Davies (67), Rees for Gravelle (41), Wyatt for Afeaki (49), A. Jones for Louw (49). Not Used: Williams, Hercus.
Glasgow: Lamont, O'Hare, Craig, Henderson, Roberts, Parks, Pinder, Murray, Lawson, Harrison, Barker, Hamilton, Dearlove, Wilson, Beattie.
Replacements: Gregor for Lamont (80), Morrison for Craig (58), Hayter for Pinder (46), Thomson for Lawson (71), Tckachuk for Harrison (41), Beveridge for Wilson (41). Not Used: Swindall.
Referee: Donal Courtney (Ireland)