The prize is Wayne Bennett’s seventh title as a coach but his first with a second club since taking over in 2009.
Again Carney missed the conversion meaning the Roosters held just a narrow 6-8 lead with a quarter of the half gone.
Leilua squeezed in at the corner for the Roosters shortly after, but referee Tony Archer disallowed the score for an earlier forward pass.
Soward missed an opportunity to level the scores when he dragged a penalty attempt wide of the post.
Despite the efforts of both sets of players neither team could improve on their score meaning the Roosters took their narrow 8-6 into the break.
Winger Jason Nightingale snatched the lead back for the Dragons by squeezing between two Roosters defenders to cross in the corner.
Soward’s conversion extended their advantage to 12-8.
Aided by the increasingly heavy rain, the Dragons scored again through Nightingale in the 60th minute.
Soward was on target with the conversion meaning the lead was stretched to ten points at 18-8.
The Dragons went further ahead when after Sean Kenny-Dowall had spilled possession, hooker Dean Young scored from close range from a neat Nathan Fein pass.
With Soward successful with the boot, their lead was extended to 26-8.
Fien sneaked over from dummy-half after the Roosters prepared for a Soward field goal and the five-eighth didn’t miss his conversion, meaning the Dragons took a unassailable 32-8 lead with less than ten minutes remaining.
The Roosters tried bravely to register a consolation score but were plagued by a series of handling errors.
Dragons fullback Darius Boyd capped a fine season by collecting the Clive Churchill medal.