A controversial first-half try from Gareth Maule in addition to three Aled Thomas penalties had put the home side comfortably in the driving seat at 16-3 at the interval.
But the Dragons refused to lie down and a converted score from winger Adam Hughes in addition to a Tom Prydie penalty reduced their arrears to 16-13.
But Stoddart, who broke the tibia and fibia in his left leg during the World Cup warm-up with England at Twickenham last year, dived over to crown his return to action.
Scarlets, coming into the clash on the back of two straight defeats in the league, dominated almost the entirety of the first half but it was the Dragons who stuck first when Steffan Jones landed a monster penalty from just inside the home team's half.
Liam Williams appeared to carry the most persistent threat for the Scarlets however and after a fine break in the opening stages he again cut free and his clever kick to the corner soon after Jones' penalty nearly yielded the first try of the match but for the quick thinking of Toby Faletau.
Conditions were playing havoc as evidenced when Jones horribly miscued a penalty kick for touch that dribbled straight back into the hands of the home side and from the subsequent passage of play the Scarlets drew level when Thomas, starting in place of the rested Rhys Priestland, slotted over a penalty.
In the 22nd minute Thomas had the chance to put the home side in front for the first time in the match but his penalty kick from a near identical position to his first drifted wide.
The first try appeared close though but no-one expected it to come in quite the fashion that it did.
Liam Williams continued to shine for the home team and his incisive break from inside his own half won his side a penalty in the 26th minute from much closer range than Thomas' last attempt.
The kick looked extremely straightforward but rather than three points the Scarlets ended up with seven in bizarre and somewhat controversial circumstances.
Thomas's penalty attempt cannoned back off the inside of the right-hand post, dropping into the left-hand corner where Maule was quickest to react and touch down.
Referee Nigel Owens went upstairs to the television match official and after lengthy TV replays it was decided the ball had not passed through the posts and the try was eventually awarded. Thomas added the extras from the left touchline to make it 10-3.
In the final ten minutes of the first half Thomas continued to thrill and frustrate from the kicking tee, slotting over two more extremely tough penalties, sandwiched between an extremely simple miss, meaning Simon Easterby's troops went in at the break deservedly 16-3 to the good.
Soon after the break the Dragons fly-half Jones missed badly with both a penalty and a drop goal attempt as time appeared to be running out for them.
But on the hour mark Prydie took over the kicking duties for the away team and reduced the deficit to 16-6 with a penalty from wide on the left.
Three minutes later and the away side were right back in it when Hughes collected Dan Evans' clever kick through to slide over in the greasy conditions, Prydie converting from out wide to make it 16-13.
But all that Dragons forward momentum was lost in the 66th minute when scrum-half Jonathan Evans was sent to the sin bin for taking the man out off the ball, replacement Priestland firing over the subsequent penalty with virtually his first touch of the ball to stretch the lead back out to 19-13.
From then on the game was up for Dragons and replacement Stoddart slid over in the left corner to seal the win.
|Scarlets Score Card|
|Newport Gwent Dragons Score Card|