After 83 minutes of stalemate, Brown pounced on a rebound to head the Shots to their first victory in six matches.
AFC Wimbledon remain in the bottom two, however, after they were denied maximum points by Adebayo Akinfenwa's equaliser for Northampton Town.
The kickoff at Kingsmeadow was delayed by 20 minutes due to floodlight failure, but when the action began, Alan Bennett put the division's bottom club ahead.
Akinfenwa's 17th goal of the season, which came on 76 minutes from the penalty spot, prevented Wimbledon following Aldershot out of the drop zone.
At the other end of the table, Rotherham United fought past in-form Bristol Rovers to climb back into the automatic promotion places.
That only proved to be a consolation, though, as Rotherham climbed to third and inflicted a first defeat in six league games on Rovers in the process.