As expected captain Paul Gallen didn't take his place in the side due to the knee injury he suffered in last weekend's loss to South Sydney.
The Sharks were boosted by having coach Shane Flanagan back in the box just five days after his operation for a twisted bowel.
Cronulla did well to go into the break 10-6 down when they repelled Melbourne time and again late in the opening 40 minutes.
Melbourne opened the scoring in the 13th minute through centre Dane Nielsen when he pounced on a dropped ball from winger Isaac Gordon, who spilled a towering Cooper Cronk bomb.
Gordon went from villain to hero within three minutes when he crashed over in the right corner after a neat backline move, with Carney bringing in fullback Matthew Wright who stretched the Storm defence before finding his winger.
The Storm hit the lead for the second time when winger Matt Duffie dived over in the left corner after a well-worked backline move.
-AAP