Maule crossed in the first half for an enterprising try after chasing up a loose ball from a missed penalty goal, while Stoddart came on at half-time to end more than year out with a broken leg, and soon went over in the left corner.
Scarlets, who stay in second place, gave a debut to South African lock Johan Snyman with George North returning to the right wing, while Thomas took over at fly-half from Rhys Priestland.
The Dragons, who have yet to win at Parc y Scarlets, were without Wales blindside Dan Lydiate after he broke an ankle in last week's win at Edinburgh.
Outside-half Steffan Jones gave the Dragons the lead on 10 minutes with a 45-metre penalty, which was cancelled out by his opposite number Aled Thomas six minutes later.
The best chance of an early try came when Scott Williams put North through but in the wet conditions the Wales wing failed to hold on.
But an incisive break by in-form full-back Liam Williams led to another penalty goal. Although Thomas' kick hit a post and failed to go over, quick-thinking Maule followed up the loose ball to score in the left corner, with Thomas adding the extras.
That lead was extended as Thomas added further three-pointers on 34 and 38 minutes as the Scarlets eased into a 16-3 advantage.
The Dragons failed to get into the opposition 22 in the first half but they responded with 10 unanswered points after the break.
Tom Prydie, who took over the kicking duties from Jones, put over a penalty to cut the arrears, and that was followed by a try for wing Adam Hughes who slid in after Williams failed to deal with Dan Evans' grubber kick. Prydie converted to cut the Scarlets lead to three points.
But the Dragons comeback was nipped in the bud when scrum-half Jonathan Evans was sin-binned for a tackle off the ball, allowing replacement fly-half Priestland to put over a penalty on 66 minutes.
And 10 minutes from full-time Stoddart ran in for Scarlets' second and decisive try.
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