Glasgow hit back through late tries by Colin Gregor and Sam Pinder, in the final seconds of normal time, to sum up the Scarlets' dismal season.
Phil Davies's men had already weathered a first-half battering before Stoddart's brilliant double as their fragile defence and ill discipline surfaced once again. The defeat now leaves the Scarlets five points behind leading Welsh side Cardiff Blues having played an extra match in the Magners League.
The Scarlets were handed a boost by the return to action of skipper Simon Easterby from a persistent knee problem. But his inclusion was dampened by the loss of Wales internationals Dwayne Peel and Mark Jones for the visit to Scotland.
The West Walians had hoped to bounce back from last weekend's defeat at the Arms Park and finish a disappointing campaign on a high note. Rock bottom and whitewashed in the Pool of death in the Heineken Cup coupled with failure to qualify for the latter stages of the EDF Energy Cup has left many Scarlets fans disgruntled.
However, their travelling army of support were boosted by the sight of bright sunshine on their arrival in the west end of Glasgow. But they faced a tough task as Sean Lineen's men had lost just once in the Magners League - to the lowly Dragons - at Firhill this season.
A scrappy opening seven minutes saw fly halves Dan Parks and Stephen Jones both guilty of missing easy penalty opportunities. After Parks had booted straight into touch from his own kick-off, the Scotland outside half had the chance to make amends but he dragged his 35 metres kick wide from in front of the posts on five minutes.
Yet he wasn't the only one to lose his radar in the bitter cold Firhill wind as two minutes later Stephen Jones also misfired. Jones regained his composure to open the scoring on 11 minutes with a close-range penalty.
But Glasgow hit back and following wing Max Evans's darting run, the home side saw their relentless pressure rewarded as the Scarlets' weak defence - a recurring theme throughout the season - creak in the 15th minute.
Parks side stepped inside Mahonri Schwalger and then converted his try. He then extended the lead with a penalty before Jones hauled the Scarlets back into the match with a penalty on 33 minutes. But Parks blasted home his third successful kick to hand the hosts a deserved 13-6 interval advantage.
The Scarlets responded and flew out of the traps in the second period with Stoddart's solo try within 78 seconds of the interval. Jones levelled the scores with the conversion. Parks missed on 52 minutes and he was made to pay two minutes later as Stoddart burst through again to score under the posts.
Jones kicked his third penalty on 61 minutes to give the Scarlets a ten-point lead before Stoddart's sin-bin. And Glasgow seized their opportunity with Gregor's try on 71 minutes before his chip and chase fell into the arms of replacement Pinder in the 80th minute.