With the game completed in near darkness due to a power failure just before kick-off, the Dragons held on for just the second win from their past five games, Trent Merrin sealing victory with a try 13 minutes from time which appeared to feature a Matt Cooper knock-on in the lead-up.
Stand-in skipper Wade Graham scored a double for the Sharks, who could drop out of the top four if South Sydney win on Monday night, but Cronulla remain without a win in Wollongong since 1989.
The Dragons made a horrendous start when Jason Nightingale misjudged the kick-off and could only watch as it bounced over the dead ball line.
The ensuing drop out was only marginally better, things going from bad to worse when Kyle Stanley - a late call-up for Jamie Soward (back spasms) - was stretchered from the field with a suspected torn ACL after just six minutes.
An eventful start was capped when Mitch Rein scooted over from short range, the Dragons rewarded for an opening in which they showed surprising endeavour with the ball.
The home side continued to chance their arm but the Sharks were gaining in confidence from their ability to repel, Graham's powerful first seeing him bust through a feeble Nathan Fien attempted tackle.
The Sharks were coming hard as halftime approached and as Todd Carney chased a lead, he threw a poor pass behind his runners, Brett Morris scooping up the crumbs and racing 75 metres to score as the hooter sounded.
Cronulla applied the squeeze to open the second stanza, Jeff Robson securing a repeat set with a neat grubber before laying on a delayed ball for Graham to spear over to leave the Sharks trailing 12-10.
Another Robson repeat set and Graham attempted to turn provider only for his ball to a runaway Anthony Tupou to be called forward, Cronulla receiving no such favour from the referees as Merrin clinched the win.