Melbourne fullback Billy Slater scored his side's only four-pointer and halfback Brodie Croft booted three goals.
The Tigers have now won four of their five matches and move into third spot on the ladder, while the Storm have won only two games and remain in seventh position.
The occasion served as a homecoming for 13 New Zealand-born players across both teams, including four ex-Warriors and former Kiwis captain Marshall among the Tigers line-up, and also for their coach, former Warriors mentor Ivan Cleary.
The Tigers dominated early but the Storm were the first to score through a Croft goal in the 17th minute, after the Tigers were fortunate not to concede a try but were penalised for interference.
The Tigers missed two scoring chances with Storm right wing Suliasi Vunivalu preventing his opposite, Malakai Watene-Zelezniak, from crossing, and left wing Josh Addo-Carr holding centre Esan Marsters up in the ingoal area.
Addo-Carr then cost his side dearly when he fielded a kick inches from the sideline and flung the ball behind him infield only for it to land in Marsters' hands before Aloiai ran in to give the Tigers a 4-2 lead in the 34th minute.
Melbourne struck back on the final play of the half, however, when playmaker Cameron Munster kicked on the first tackle for Slater to win the race and slide over out wide. Croft's conversion sent them to the sheds four-points in front.
Another Croft penalty increased the Storm's lead near the end of the third quarter, but they failed to retrieve the kick-off to gift the Tigers attacking possession, before Luke Brooks put Lawrence over for their second four-pointer with Lolohea's conversion levelling the scores.
Croft then missed a long-range shot at breaking the deadlock, before Marshall calmly stepped up and banged the ball between the uprights.
The Storm threatened to steal the result with Hoffman galloping over before being called back for a forward pass from dummy-half, which summed up a frustrating night for the reigning premiers.
Wests Tigers 11 (Josh Aloiai, Chris Lawrence tries, Tui Lolohea con, Benji Marshall field goal)
Storm 10 (Billy Slater try, Brodie Croft con, 2 pen)