Twice Johnson threw passes that missed wing Jonathan Wright, the first a no-look speculator that flew into touch and the second floating two metres forward, when a chest ball on either occasion could have reaped points.
A third chance evaded the Warriors when Vatuvei couldn't handle a Matulino tip-on, but they clung on defensively, as the Storm began applying pressure, with Vatuvei and Bodene Thompson both producing try-saving tackles.
But after absorbing plenty of pressure the Storm then capitalised when right wing Mahe Fonua cleaned up a loose pass from Will Chambers to dive over out wide, with Cameron Smith converting to make it 6-0 just inside the second quarter.
The Warriors' woes continued with more errors from Hurrell and Leuluai coming early in the tackle count, and Johnson committing a third mistake when he lost possession after a desperate Billy Slater ankle tap, when the No7 looked to be setting sail for the try line.
Soon after the Storm again came out of nowhere to add their second four-pointer, with left wing Marika Koroibete brushing past Wright and Hurrell to race in, with Smith again adding the extra two.
The Warriors received a boost three minutes before halftime when Vatuvei powered through to ground the ball on the chalk just inside the touch line, despite the attention of four defenders, but Johnson was unable to convert.
Melbourne extended their lead seven minutes after the resumption with Cooper Cronk and Slater working a brilliant chip and regather before the No 7 and Kevin Proctor handled to send Chambers over.
The Warriors kept themselves in the contest by earning a repeat set in the lead up to Nathan Friend ducking over from dummy-half with a quarter remaining, but they were quickly put on the back foot when Koroibete broke downfield before Slater put Tim Glasby in for their reply.
With the clock winding down, Sebastine Ikahihifo wrestled his way over to keep hopes of a late Warriors rally alive, but the Storm again kicked clear when Koroibete dived in to score the match-winner.
Storm 30 (Mahe Fonua, Marika Koroibete 2, Will Chambers, Tim Glasby tries; Cameron Smith 5 cons)
Warriors 14 (Manu Vatuvei, Nathan Friend, Sebastine Ikahihifo tries; Johnson 1 con).