Samir Carruthers came close to making the breakthrough for the Dons in the first half, with his strike from distance coming back off the bar.
Just after the half-hour mark, Crewe won a free-kick in a promising position after Stephen Gleeson brought down the lively Chuks Aneke, before the Englishman brushed himself down and curled home the set-piece past David Martin to give the home side the lead.
Shaun Williams stepped up to take the penalty, but his effort was saved by Ben Garrett in the Alexandra goal.
The away side would live to rue the missed opportunity, with AJ Leitch-Smith doubling the Railwaymen's advantage after racing onto Anthony Kay's block and slotting past the keeper.
The Dons tried to get back into the game, but failed to make the most of their opportunities, and they are now eight points off the playoff places.