Down by 27 points early in the final quarter on Friday, the Demons stormed home with the final four goals of the match in an extraordinary late surge.
Melksham marked about 40 metres out on a slight angle with a minute to go, calmly going back and popping his shot over the line.
"I'm rapt that I put it through because I kicked three out on the full in the first quarter so it was good payback," Melksham told the Seven Network.
Brisbane cleared it into their forward-50 in the dying seconds as Eric Hipwood tried to claim a mark 50m out, but the umpires decided Jake Lever had some of the ball and the game ended.
Melbourne prevailed 16.9 (105) to 16.8 (104) to move just a game behind the third-placed Lions in a crucial result for both teams' premiership prospects.
Coming into Friday night's match with 12 straight regular-season losses at the grand final venue dating back to 2015, the shell-shocked Lions conceded the first four goals of the game in the opening five minutes.
After surviving the early carnage, led by red-hot Demons star Christian Petracca, the Lions settled to dominate the second and third terms by booting 12.4 to 6.2.
Leading by 22 points at the final change, Brisbane looked to be home, but the Demons made an unlikely late run as the Lions' midfield dominance stopped.
Chris Fagan has constantly bristled at questions why his team struggles at the Melbourne colosseum, but the Lions have won there just once in their past 15 visits.
"It's hard to cop because we did a lot right, but at the end of the day I'd say we didn't deserve to win, necessarily," Fagan said.
"We'll take learnings from it and we'll go and have a look at the last five minutes and how we handled that.
"Melbourne are a good footy side and when you play them here, you've got to do a lot of things right to win."
The Demons racked up their biggest score in nine weeks as their under-fire forward line functioned better.
Petracca starred with his second four-goal haul in as many weeks, creating problems for the Lions' defence in a more unpredictable Demons forward-half.
"It just shows the character and fight of our footy team; for large parts of that game, especially through the second and third quarter, you could well and truly tell that Brisbane were on top," Melbourne coach Simon Goodwin said.
"For our guys just to stay present, stay in the now and then to adjust and work their way back - with seven minutes to go we're still 24 points down - to get a win, it's incredible."
Melbourne defender Harrison Petty was eventually substituted out in the third quarter after struggling with a back injury, replaced by Joel Smith.
The Demons went in with captain Max Gawn as their main ruckman after opting to axe star recruit Brodie Grundy as the Demons hope he can hone his forward craft in the VFL.
Melbourne host Adelaide at the MCG next Sunday, while the Lions are back at the Gabba to face reigning premiers Geelong.
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