Bamford's brace plus goals from Dele Alli and Shaun Williams handed the Dons a comfortable victory, with Darren Potter's own goal the only blot on their copybook.
Dons tore Stevenage apart in the first half as they raced into a three-nil lead inside the opening 23 minutes.
Bamford netted the first two goals, glancing a header in off a post from Williams' free-kick and then sweeping home Luke Chadwick's cut back for his eighth goal of the season.
Alli added a third when he slid in to control Chadwick's overhit throughball and produced a fine improvised finish to poke home his first League goal while still grounded.
Boro pulled one back three minutes before the break when Potter turned a fizzing cross from Luke Freeman past his own goalkeeper under pressure from Roarie Deacon.
But Williams put the Dons 4-1 up from the penalty spot after 62 minutes, sending Chris Day the wrong way after Sam Wedgbury had tripped Alli in the box.
Stevenage rallied late on as Freeman drilled a shot against a post and substitute Oumare Tounkara slotted wide when one-on-one with David Martin but Dons ran out convincing winners.