On-loan Chelsea striker Bamford nipped in at the near post to poke home the only goal of the game in the 20th minute to cap an impressive personal display.
The hosts dominated the first half and Bamford put them ahead when he tapped in Jason Banton's low cross after the on-loan Crystal Palace winger had flown past Kelvin Mellor down the left.
Bamford had earlier dinked an effort on to the top of the crossbar, while Shaun Williams saw his fierce long-range strike well saved by Crewe goalkeeper Alan Martin and then also curled a free-kick inches past a post.
Darren Potter looked set to double the Dons' lead from close range shortly after the break but he was thwarted by an excellent last-ditch block by Mellor.
Crewe improved in the closing stages and George Ray wasted a golden opportunity when he planted a free header wide from Ajay Leitch-Smith's free-kick, while fellow defender Matt Tootle was twice narrowly off target.
Dons 'keeper David Martin then produced an excellent reflex save at the death to tip over Ryan Colclough's close-range volley as the hosts hung on.