The first try was scored by Felise Kaufusi after the frantic Storm sent the ball through 13 pairs of hands on the final tackle.
Billy Slater then crossed for their second in the 50th minute after speedster Josh Addo-Carr kicked a ball through into the in-goal for the veteran fullback.
With those tries converted by Smith, Melbourne opened up a 16-10 lead.
After a successful Smith penalty for the Storm to go ahead 18-10, the Eels were then back in the fight when winger Semi Radradra scored in the 66th minute to set up a thrilling finale.
Melbourne were quick out of the blocks with Addo-Carr touching down in the fourth minute, after centre Curtis Scott was able to get the better of his opposite Michael Jennings.
But the home side lost their way with the sin-binning of Cameron Munster in the 17th minute for a professional foul after tackling Kenny Edwards from an offside position.
The Eels scored two tries while Melbourne were a man down to set up their six- point halftime lead.
They first caught the Storm short out wide with Kirisome Auva'a crossing untouched, while their second came in the 31st minute when halfback Mitchell Moses flicked a no-look pass for exciting young fullback Will Smith to run through a hole.
Parramatta had two players - Manu Ma'u and Daniel Alvaro - both put on report which could hinder their finals hopes.
- AAP