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Ellison gave the Shrimps a dream start in the League Two clash with a superb opener after eight minutes. The summer signing received the ball 30 yards out from Danny Carlton's lay off and let fly with a left-foot volley from 30 yards that flew into the top left-hand corner of Jamie Young's goal.
Substitute Alessandra then joined in the act with a finish from a similar distance 13 minutes from time.
As Aldershot threatened to fight their way back into the game, Alessandra produced a fierce volley to beat Young to his right to seal the points.
Away from the goals Aldershot had a great deal of possession and went close when Danny Hylton hit the crossbar after 20 minutes with a dipping shot from the edge of the area that took a deflection.
Peter Vincenti thought he had levelled the scores soon afterwards but the goal was ruled out for a push.
The visitors' best chance in the second half came when Michael Rankine missed the target from four yards out when he looked certain to score.
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