Melksham and Christian Petracca (31 disposals, one goal) were key figures in the Demons' 18.11 (119) to 12.11 (83) victory in front of 35,844 fans at the MCG on Sunday.
It was Melbourne's first win since thumping Sydney at the same venue on May 25, at the end of their mid-season form spike.
The Demons (6-11) now sit 13th, while North (4-12-1) are third-from-bottom.
But the Dees' drought-breaking win was soured when Tom Sparrow was knocked out late in the match by a stray arm from Kangaroos ruckman Tristan Xerri.
Play was held up for several minutes while Sparrow was assessed by medical staff and eventually taken off the ground on a stretcher.
Xerri tried to check on Sparrow before he was driven off but was pushed away by Melbourne players, who repeatedly went after the ruckman in the dying stages.
The incident was a flashpoint in a physical affair littered with spotfires.
Melksham also looked in trouble after high contact from North defender Wil Dawson left him nursing a sore head and shoulder before half-time.
But the 33-year-old passed a head injury assessment and added four goals after halftime as the Demons resisted a Cam Zurhaar-led surge by the Kangaroos.
Zurhaar kicked three goals in the third quarter, dragging the Kangaroos within three points, before Melbourne steadied and finished full of running.
Christian Salem (25 touches), Harvey Langford (24) and Clayton Oliver (21) were influential in a largely scrappy affair, while Bayley Fritsch kicked three goals.
The final margin made for a 95-point turnaround from the last time the teams met, when the Kangaroos were comprehensive winners in round two.
North were well-served by Colby McKercher (29 disposals), Harry Sheezel (29) and returning veteran Luke Parker (27).
Cooper Harvey, son of club legend Brent, kicked four goals and Jack Darling matched Zurhaar's contribution with three.
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