The visitors got off to a marvellous start when Wayne Curtis put them in front within three minutes. Garry Hunter picked him out on the edge of the Fleet box and Curtis had time and space to chest the ball down and blast past Paul Wilkerson.
On 23 minutes Morecambe doubled their lead when a foul by Andrew Drury on the Fleet right-wing led to a free-kick which was taken by Hunter. Chris Blackburn got a head to the ball and flicked on for Darren Kempson to score his first goal for the Shrimps.
Lee Protheroe dragged Gravesend back into the game with a belting goal direct from a 25-yard free-kick in the 39th minute.
With 63 minutes on the clock, Curtis robbed Nathan Bunce and with Michael Twiss waiting to finish from 10 yards, Wilkerson snuffed out Curtis' pass.
A 40-yard lob from Drury on the right hand touchline had Robinson scurrying back to cover as the ball floated narrowly over the bar on 72 minutes.
With six minutes remaining a Protheroe volley crashed into the side netting and in the final minutes both Jean Michel Sigere and Bunce had headers cleared from under the Morecambe crossbar with Robinson beaten.