Saturday's victory - the Dragons' first of the campaign - was a dour and ugly affair which hung in the balance until the final quarter of an hour.
Interchange hooker Jacob Liddle burrowed his way over the line in the 69th minute to send the rain-soaked crowd of 6211 at Kogarah Oval into a frenzy.
The Dragons had let close games slip from their grasp in the first two rounds of the season, falling to Canterbury and South Sydney before last week's bye.
Their lack of game control in the clutch moments led to coach Shane Flanagan giving halfback Lachlan Ilias a public dressing down.
Ilias was solid on Saturday but Liddle showed great cunning to spot a gap in the Melbourne line and dart over to take the Red V past an unconvincing Storm.
"'Lids' comes on and gives us that bit of speed out of dummy half and another threat," Flanagan said.
"It's really important, they (starting hooker Damien Cook and Liddle) are a tag team.
"'Lids' had a bad preseason, three hand surgeries, didn't train a lot, so he's coming good at the right time."
Without halfback Jahrome Hughes, Melbourne struggled for fluency in attack and their all-star spine struggled to hit top gear.
Craig Bellamy's side had 46 play-the-balls in the Dragons' 20m zone but their sole try was scored by Ryan Papenhuyzen just before halftime.
"We had a couple of chances that we could have and should have taken - and we paid the price for it," Bellamy said.
Papenhuyzen's four-pointer came after stand-in Storm halfback Tyran Wishart had kicked a penalty goal to open the scoring for the visitors.
Valentine Holmes crossed for the first try of the afternoon in the 34th minute before Papenhuyzen gave Melbourne an 8-6 halftime lead.
Melbourne had chances to add more in the second half but Shawn Blore and Eli Katoa both fumbled over the line as Saints' desperation shone through.
"We're a good footy side, but we can't put ourselves under pressure like we did today," Flanagan said.
"40-odd tackles on our try-line, we need to be smarter."
After missing an earlier penalty goal from right in front, Holmes drew the Dragons level in the 62nd minute with a straightforward kick to send the game down to the wire.
Liddle was the hero from close range in the 67th minute as the hooker shrugged off some lax Storm defending to give his side the lead.
Holmes added the extras to give the Dragons a six-point buffer which Melbourne could not overcome.
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