The Rabbitohs worked their way to a vital 14-0 halftime lead and continued to defend exceptionally in the second half.
It wasnt until the 55th minute that the Storm finally breached South Sydneys line via a Billy Slater try.
At 6-14 down Melbourne was storming home, even more so when Slater seemed to have scored a fair try soon after only to have it rubbed out by the video ref; it was deemed Nathan Merritt had been taken out in the air by Will Chambers when trying to defuse a bomb.
Instead of the score-line reading 12-14 with the likely conversion from a handy angle, South Sydney had a penalty and headed straight downfield to create what proved to be a match-clinching try for man of the match Issac Luke.
Chambers scored in the 72nd minute but the Storm had run out of time to fashion an escape and the Rabbitohs were in no mood to give them a glimmer of hope.
South Sydney goes through to week three while Melbourne will be at home next weekend facing the winner of Sundays elimination final between (6) Canterbury-Bankstown and (7) Newcastle.
At ANZ Stadium, Homebush
(2) South Sydney Rabbitohs 20 (Dylan Farrell, Jeff Lima, Issac Luke tries; Adam Reynolds 3 conversions, penalty).
(3) Melbourne Storm (Billy Slater, Will Chambers tries; Cameron Smith conversion).
Halftime: 14-0 Rabbitohs.
Referees: Shayne Hayne and Jared Maxwell.