Brown's air shot from a backpass in the 38th minute gave James Dunne a present and it also gave the League Two side a mountain they were unable to climb.
Stevenage went ahead after 14 minutes when Darius Charles found Luke Freeman in space down the left and his low cross was turned into his own net by Dons defender Pim Balkestein.
Boro were gifted a second when Brown missed Mat Mitchel-King's looping back pass and Dunne could not believe his luck as he rolled into an empty net.
The visitors pulled one back when Brendan Kiernan struck but Stevenage captain Mark Roberts restored the two-goal advantage on the stroke of half-time by ensuring Marcus Haber's header crossed the line.
In a quieter second half, Freeman had an effort saved at the near post by Brown before the winger blazed wastefully over the bar with 15 minutes left.