Second-half strikes from Barry Corr and Britt Assombalonga took the Shrimpers up to 10th in the league, leaving them four points away from the play-offs with just two games left to play.
Brown said: "It feels magnificent to have won. The team played well on Saturday (against Bristol Rovers) and again, with the exception of a couple of errors, I thought we were comfortable winners here.
"The players have shown me what they're capable of in terms of power and energy, the difficult side of the game.
"We were up against a little bit of a wind but that died down in the second half and our centre-halves spent an awful long time facing their own goal and we did not cope well with that at Rochdale.
"The Aldershot players are game. They are willing and they had four strikers on towards the end, so I'm delighted for our young goalkeeper (Dan Bentley).
"He's had two games now under me and kept two clean sheets and you cannot ask for more than that, but when the back four work as hard as that then you have to be happy as a manager."