Ahead 16-6 with less than 20 minutes remaining, the Sea Eagles' lead was wiped out following two Xavier Coates tries.
But with two minutes left and scores tied, Storm captain Harry Grant was penalised for making contact with the legs of Luke Brooks while the Manly ace was attempting a field goal.
Reuben Garrick made no mistake with his kick from directly in front to put the visitors two points up.
In their first win in Melbourne since 2019, the Sea Eagles capitalised on a slow, sloppy start from the Storm to set up their third-straight victory.
For just the second time this season, Melbourne conceded more than 12 points in the first half as Manly went to the break up 16-6.
Storm coach Craig Bellamy unleashed a fearsome halftime spray on his players following the slow start.
Their errors in the first half were completely uncharacteristic, frustrating Bellamy as he marched out of the coach's box multiple times.
Manly travelled south up for the fight.
Just minutes after coming on in the first half, Sea Eagle Jake Simpkin was placed on report for a dangerous tackle on Grant Anderson.
Simpkin avoided being sin-binned, but Anderson was forced from the field for a HIA.
Melbourne were out for retribution, with enforcer Nelson Asofa-Solomona throwing his elbow back in a tackle from Simpkin, collecting him in the head.
Asofa-Solomona's rush of blood proved costly, as it came in the same play captain Harry Grant crossed for a try just before halftime.
But the four-pointer was overturned, leading Bellamy furious.
Coates was the shining light in Melbourne's first home loss of the season, finishing with the ninth hat-trick of his career.
The result moved Manly to 9-8 this season and back into the top eight ahead of another blockbuster clash next Sunday with Canterbury.
The Storm have just five days to back up before facing the Sydney Roosters at Allianz Stadium.
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