It proved to be the only score of the opening period with both sides playing their part in an absorbing encounter, Sharks skipper Paul Gallen leading the way with another tireless performance as he clocked up 223 running metres and made 39 tackles.
In the second period, the Dragons came racing out of the blocks but the Sharks withstood wave after wave of attack, with wingers Matthew Wright and John Williams defusing a number of bombs from Soward to frustrate the visitors.
The longer the Sharks held on, the less likely the Dragons looked like scoring, and a brilliant 70-metre break from prop Andrew Fifita relieved the pressure.
Shortly after, Shane Flanagan's side scored their second four-pointer through Wade Graham in the 57th minute.
Carney brilliantly kept the ball alive on the final tackle with a popped pass to Jeff Robson, and the halfback's perfect grubber kick into the in-goal area was pounced on by the former Penrith utility to the delight of the home supporters.
The Dragons thought they had pulled a try back from Jason Nightingale in the 70th minute, but his left foot was adjudged to have touched the line.
CRONULLA 12 (W Graham J Morris tries T Carney 2 goals) bt ST GEORGE ILLAWARRA 0 at Toyota Stadium. Referee: Matt Cecchin, Adam Devcich. Crowd: 21,380.