They took the lead through Paul Burnett in the eighth minute but Leon Antoine's glancing header from a corner quickly levelled the scores.
There was then a great deal of frustration for the home side, who were always on top but could not find a way past defiant keeper Fred Howe.
Two substitutions eventually tipped the scales. Petrit Elbi had only been in the action for five minutes when he restored Aveley's lead in the 79th minute and fellow-sub Shane Oakley quickly added another, a fine solo effort.
Elbi added the fourth, the tenth goal conceded by Sudbury in three days. It is not going according to plan for the pre-season title favourites.
Pictured: the grounded Howe and Aveley's Billy Holland (8) look on as Elbi moves in to score his first goal. Pic by Mikey Cartwright.
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