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After dominating the first half the home team fell behind to Scott Doe's header on the hour mark before Brown headed in with four minutes as the Daggers extended their unbeaten run to seven games.
Aldershot were the first to threaten through Jordan Roberts, on his league debut, and then Wilko Risser, who missed a gilt-edged chance when he fluffed his kick straight in front of goal.
They continued to spurn presentable opportunities in the first half as Adam Mekki and Adam Vincenti failed to get their heads on target.
Dagenham's Michael Spillane brought out a spectacular save from Jamie Young on 51 minutes before Danny Hylton blazed over from eight yards for the hosts six minutes later.
But the deadlock was broken when Doe headed in Damien McCrory's corner.
Dagenham could not hang on, however, and Brown rose high to head in Mekki's cross on 86 minutes.
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