Storm winger Josh Addo-Carr scored a try in the 66th minute to give his side the lead for the first time, while Felice Kaufusi capped off the 20-14 victory when he barged over as the clock wound down.
It was Melbourne's seventh win in eight games this season, moving them two points clear of the chasing pack.
Last year the Warriors were trounced 42-0 in the same match but showed their intent from the opening whistle and David Fusitu'a collected his first try in the 10th minute.
