His NSW rival Jarryd Hayne was comparatively kept quiet.
On the back of the win, Craig Bellamy's team climbed into the top eight and went level with the Eels on 18 points.
Melbourne made a flying start and were up 18-0 after just 23 minutes.
Slater threw the final pass to winger Sisa Waqa for their opening try and then had points against his own name after some good lead-up work by five-eighth Ben Hampton.
Within five minutes, he was in again after a Hampton bomb fell back off some leaping contenders into his arms.
Parramatta finally had some possession and field position and started to threaten the tryline.
Centre Ryan Morgan, who replaced injured Blues winger Will Hopoate in the Eels' line-up, showed his class in a 30-metre run as he beat Slater to touch down.
The home side then lost concentration and, with only seconds remaining in the half, allowed Parramatta in again with five-eighth Corey Norman setting it up with a monster cross-field pass to trail 18-10.
The Storm started the second half as they did the first with Slater kicking into the in-goal for Waqa.
The Eels' Vai Toutai then crossed before Sandow's sin-binning stalled their momentum with Kurt Mann scoring a double and prop Bryan Norrie also touching down.
That put the score out to 40-16 before some late efforts by both sides kept the scoreboard ticking over.
- AAP
