The on-loan Chelsea striker set up Luke Chadwick's opener and then sealed the win himself in the 90th minute after Shaun Williams had netted a brilliant free-kick, while Callum McGregor replied for the winless Magpies.
Chadwick handed the Dons a dream start after just five minutes when he neatly swept home Bamford's cut-back for his first goal of the season.
Marcus Haber almost levelled for County on his debut but his powerful header was acrobatically clawed away at the second attempt by goalkeeper David Martin.
But the Dons were firmly in charge and Williams curled a fine 25-yard free-kick into the top corner of the net to double their lead after 37 minutes, while Bartosz Bialkowski denied Bamford in a one-on-one situation and Stephen Gleeson saw a header hit the post.
County improved after the break and McGregor handed them a lifeline when he raced clear to slot the ball past Martin in the 76th minute.
However, it was the Dons who finished the strongest and substitute Izale McLeod missed two late chances before Bamford rammed in a loose ball at the death for his sixth goal of the season.