The Giants appeared destined to fall short with about two minutes left in the final quarter in Alice Springs when Kelly marked, played on and launched the winner from close to 60m out.
GWS held on for a dramatic 7.5 (47) to 5.15 (45) victory that puts the Giants (28 points) four points outside the eight and leaves the Demons' top-four spot vulnerable.
"I can't question the players' effort, I can't question their want, their desire, really tough conditions - but to have 27 shots at goal, 70 entries, plus 40-odd in the contest and lose a game of footy, that's tough," coach Simon Goodwin said.
"Inefficiency cost us again and it's something we need to get right
"We're working incredibly hard on it but it's just not coming together ... if we keep that effort and we keep that work, it'll turn pretty quickly.
"We weren't able to take opportunities when we had them and our inefficiency cost us the game for a few weeks in a row now.
"So we need to get that right as a footy club ... we aint stopping in our bid to get better, we're gonna keep working on it and eventually the wheel will turn."
Melbourne kicked 8.18 in their loss to Geelong, 8.18 in their win over Collingwood and 8.13 when they beat Carlton.
The loss, Melbourne's second on the bounce, leaves the fourth-placed Demons (36 points) eight points behind third-placed Brisbane and only ahead of the fifth-placed Western Bulldogs on percentage.
Star forward Bayley Fritsch will go for scans after he had his foot stepped on and was promptly substituted for Jake Melksham in the first quarter.
GWS coach Adam Kingsley wouldn't buy into the idea the Giants had got out of jail, instead hailing his defence.
"The stats will say they were really dominant. I thought they had really good patches," Kingsley said.
"But I felt like we had our moments too.
"For us to be able to fight, keep strong behind the ball particularly, against a lot of entries ... I was really pleased with our performance."
Giants Tom Green (38 disposals, 16 contested possessions), Kelly (26, two goals) and Stephen Coniglio (30) had plenty of the footy.
Melbourne's Jack Viney racked up 40 disposals, eight clearances and 607 metres gained and was well-supported by a wasteful Christian Petracca (34 disposals, 556 metres gained).
Petracca kicked four behinds in the opening half while Toby Greene booted two second-quarter goals to keep scores level at halftime.
GWS came out of the main break with greater intensity and nudged out to an 18-point lead midway through the third quarter.
Melbourne fought back and Ed Langdon restored their lead early in the fourth quarter.
GWS snatched victory when Kelly let loose with the decisive long bomb.
"It was a wonderful goal," Kingsley said.
"Even the decision to 'I'm going to take this on myself' and finish - that tells you enough about him".
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