The Storm faltered during the first half to allow the Rabbitohs to level 10-10 before 14 unanswered points sealed the result.
As in their heart-breaking round-one loss to the Roosters, the Rabbitohs toiled hard but again finished empty-handed.
An error by Slater led to one of South's tries but he more than compensated for that as he set up two Storm four-pointers as well as the two of his own.
His first try came in the 65th minute after a set play with halfback Cooper Cronk wrapping around Ryan Hoffman and Slater scooping up a long pass to dive across in the corner by the narrowest of margins.
Off-contract centre Dane Nielsen gave rival clubs plenty to think about with a try in each half while setting up Slater's final try with a neat grubber into the in-goal.
There was plenty on the line in front of 15,872 fans, apart from two competition points, with three former Storm players - Greg Inglis, Michael Crocker and Matt King - in the Rabbitohs line-up.
Their coach Michael Maguire is a former Melbourne assistant.
But the former Melbourne players were kept relatively quiet as the apprentice was unable to topple master coach Craig Bellamy, whose 10 years in the Storm post was celebrated in a video montage at halftime.
Melbourne's opening 20 minutes were dominant with Nielsen and winger Matt Duffie both scoring, Slater assisting in both.
The Storm looked like they would race away but Souths winger Chris Macqueen got some revenge in the 29th minute, the ball arriving via Inglis from a magic offload by powerhouse secondrower Dave Taylor.
Isaac Luke came of the bench to spark his side and he found rampaging forward Sam Burgess, who surged across under the uprights for the 10-10 scoreline.
But that was as good as it got for the Rabbitohs, the difference being the Storm's desperate defence and Slater.
Maguire lamented a five-minute period in the second half when winger Feluti Talanoa was denied a try by video referee Chris Ward, who then awarded Slater his to break the 10-10 deadlock.
He felt his side had the momentum at halftime.
"We didn't take our opportunities or we didn't hang on to the ball, that was probably our biggest issue,'' he said.
"The biggest difference between Melbourne and us is when they do pull the trigger on their plays they do execute them very well.
"There were times when we were doing that but across the whole game we weren't.
Storm coach Bellamy said the match was a replica of their round one win over Canberra, when they started and finished well but were poor in the middle.
"We got really sloppy with our handling and that let them back in,'' he said.
"I thought we were a bit dusty after halftime but I thought our defence improved then and I was real happy with how we finished the game in the last 20 minutes.
"I thought we were pretty much in control.''
- AAP
