DTH van der Merwe touched down twice for the Scots and Tom Ryder added a late third, while Ruaridh Jackson finished with a 14-point tally.
The result maintains the Scarlets' push for a Magners League play-off spot. As things stand ahead of Saturday's two games, they are sixth in the table (57 points) and within striking distance of third-placed Leinster (61).
But the Scarlets are assured of finishing the league as one of the top three Welsh sides and taking their place in next season's Heineken Cup. The Newport Gwent Dragons are out of the running after their defeat to Edinburgh.
Both Hefin O'Hare and van der Merwe looked lively in the opening exchanges at Firhill, and a break from O'Hare down the right touchline led to the game's first score.
Fly-half Jackson knocked over a sixth minute penalty after the Scarlets were adjudged to be offside at the ensuing breakdown.
The Warriors were looking to atone for the defeat to Aironi last weekend and Jackson was soon on the scoreboard again, his ninth minute penalty giving the hosts a 6-0 advantage.
However, the Scarlets responded in the 16th minute with a try from their first real attack. Centre Gareth Maule bounced off an attempted tackle from van der Merwe before feeding Stoddart who dived over and Jones converted.
The lead was shortlived, nonetheless, as van der Merwe made up for his earlier mistake by crossing for a score.
The Canadian international made ground after a strong break from midfield and linked up with O'Hare before taking the return pass and squeezing over despite the best efforts of the covering Scarlets defence.
Jones pushed a penalty attempt wide of the target shortly afterwards, before the game really sparked into life.
A scuffle between the fit-again Chris Cusiter and Scarlets lock Lou Reed developed into a melee involving players from both sides, with Cusiter and opposing captain Matthew Rees getting embroiled in a wrestling match.
Rees was sent to the sin-bin for his involvement in the incident, but Jackson was unable to convert the penalty that followed.
The Warriors were made to pay for their profligacy, Jackson failing with a further penalty attempt and then the Scarlets claimed their second try on the stroke of half-time.
With decoy runners leaving the home defence stretched, the Scarlets created an overlap which Stoddart expertly exploited to burst through the final tackle and crash over. Jones failed with the conversion attempt, leaving the Welsh region with a 12-11 buffer.
The second half began in ideal fashion for the hosts. The ever-ready van der Merwe capitalised on a turnover in midfield and some strong running from the impressive James Eddie to touch down in the corner, evading a last-ditch tackle from Stoddart.
Television match official Jim Yuille rubberstamped the score - van der Merwe's ninth try in this season's league. Jackson landed the conversion from a tight angle near the touchline.
The sides were soon level again, though, as two Jones penalties cancelled out the Warriors' advantage, the latter resulting from a cynical trip from Moray Low which earned a yellow card for the Scotland prop.
Glasgow soon edged in front courtesy of a Jackson penalty but a strong drive from the Scarlets pack, only moments later, allowed number 8 Morgan power his way over the line as the lead changed hands once again.
With their full compliment restored thanks to Low's return, the home side were able to reassert themselves at the breakdown to great effect, the Scarlets conceding a penalty in midfield which Jackson duly sent over.
But the visitors again came back, regaining possession from the restart and sparking a forward rumble which culminated in Reed bundling over from close range in the 63rd minute. Jones converted for a 30-24 lead.
Busy scrum half Knoyle touched down seven minutes later, giving the Scarlets some breathing space along with Jones' extras.
Ryder's try gave Glasgow some late hope, but it was swiftly extinguished as replacement fly-half Duncan Weir pushed his conversion attempt wide to end any thoughts of a late comeback.
This thrilling triumph completed the Scarlets' first season's double over the Warriors since the 2003/04 season, with Glasgow now destined to finish in the bottom three.
|Glasgow Warriors Score Card|
|DTH van der Merwe||2||10|
|Scarlets Score Card|