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Melbourne opened the scoring early on through centre Will Chambers and the Warriors hit back through centre Gerard Beale before wing Suliasi Vunivalu crossed again for the visitors to make it 8-6 at halftime.
The match hung in the balance throughout the second half but two penalty goals to Storm captain Cameron Smith gave his side further breathing space and the visitors held on for a hard-fought victory in front of 17,695 fans at Mt Smart Stadium.
The Warriors had their chances throughout the match but poor execution and terrific Melbourne defence saw them unable to conjure a comeback victory as they slumped to their fourth straight home defeat.
With regular five-eighth Blake Green out with a rib injury, Warriors halfback Shaun Johnson stepped up to steer the side around but his best efforts weren't enough for the home side.
The result leaves the Auckland outfit in eighth spot on the premiership ladder while Melbourne's seventh straight win saw the reigning premiers claim top spot above the Dragons and Rabbitohs.
The Warriors are left to sweat on the fitness of Luke after he was forced from the field with a calf injury in the 51st minute, and they will hope both he and Green will be fit to return for next Sunday's away game against the Titans.
Warriors lock Adam Blair also faces a nervous wait and may find himself in hot water with the NRL's match review committee after he was penalised and placed on report for a late and high tackle on Storm fullback Billy Slater inside the last 10 minutes.
The Storm enjoyed a perfect start with rookie No 7 Jahrome Hughes' flat pass putting Chambers through a hole near halfway before he beat Warriors captain Roger Tuivasa-Sheck to score in the third minute.
The Warriors were quick to reply from a penalty with Johnson's cross-field kick finding a flying David Fusitu'a who tapped it back for Beale to dive over, with Johnson's conversion giving them a narrow lead.
Errors saw the Warriors unable to get to their kicks on two sets while poor execution saw scoring chances missed on either flank, before the Storm made them pay with Vunivalu claiming Munster's chip-kick to score their second for a two-point advantage at halftime.
A double movement saw Warriors centre Solomone Kata denied a try early in the second half and the enterprising tactic of short drop-outs eventually cost them with the ball going out on the full. The resulting penalty allowed Smith to extend his side's lead.
The Warriors pushed hard inside the dying stages but Melbourne's defence held firm and Smith's second goal sealed matters in the 77th minute.
Storm 12 (Will Chambers, Suliasi Vunivalu tries; Cameron Smith 0/2 cons, 2 pens)
Warriors 6 (Gerard Beale try; Shaun Johnson 1/1 con, pens)