The win moves the Dragons into equal eighth spot on the NRL ladder, keeping their finals destiny well and truly in their own hands in front of an impressed crowd of 11,477.
But alarm bells will certainly be ringing at the Storm, who have suffered their first five-game losing streak since 2002 - the year before Craig Bellamy took control at the club.
The Storm welcomed back fullback Billy Slater from a knee injury but even that wasn't enough to stop their slide from premiership favouritism.
Less than a week after getting knocked senseless in the brutal shoulder charge that earned South Sydney's Greg Inglis a three-match suspension, Dean Young was fantastic for the home side, while Nathan Fien was tremendous filling in for the injured five-eighth Jamie Soward (back spasms).
Slater was less impressive, throwing an intercept that resulted in a 95-metre run for Morris's first try, which opened the scoring in the seventh minute.
His second was a similar effort, a 90-metre intercept in which he benefited from a mistake from one of Melbourne's other superstars - this time halfback Cooper Cronk.
The Dragons led 18-0, following tries to Morris, Beau Scott and Trent Merrin, with five minutes left before halftime when the Storm hit back in emphatic fashion.
Back-rower Kevin Proctor and centre Will Chambers punched holes in the Dragons left-handside defence to reduce the halftime lead to just six.
But Morris erupted to start the second half, scoring the second and third of his tries for the night to take his tally to 26 in 23 games at the Wollongong venue.
It was the end of their point-scoring for the night, but was still their biggest offensive output since their Anzac Day victory over the Sydney Roosters in round eight.
The Dragons managed to cling on for the final half an hour, producing some solid defensive displays the Storm were only able to breach their tryline through a scrappy try to lock Ryan Hinchcliffe.
Further souring the night for the Storm, Proctor was placed on report for a lifting tackle while Cronk could come under the microscope for a chicken wing tackle on Ben Creagh in the 47th minute.
Young, who was not officially cleared to play until after he completed Thursday's final training session, said his place in the lineup was never in doubt.
"I was always going to play ... our season was on the line tonight so there was no decision to be made really," Young said after the match.
He would not be drawn on the debate over whether the shoulder charge needs to be banned.
"My view is I get paid to play footy not to make decisions about whether or not the shoulder charge should be allowed in the game of rugby league," Young said.
"That's why I didn't come out and say anything ... smarter people than me make those decisions."
Dragons coach Steve Price praised Young's courageous performance.
"He's playing on one leg (and) he's the heart and soul of this footy club," Price said.
Captain Ben Hornby added: "He's huge ... he's the glue in our forward pack."
Melbourne coach Craig Bellamy was at a loss when asked how he plans to halt his side's worrying slide.
"We'll go back to the drawing board, and we're not giving up here," he said.
"That's probably a bit of foreign land for us (the slump) but we're pretty much sure what we need to do, it's just a matter of doing it consistently for 80 minutes."