Alex striker Mathias Pogba cancelled out Patrick Bamford's opener for the Dons and then struck the winner before visiting defender Anthony Kay was sent off for a second yellow card.
Crewe, unbeaten in their last three games, are nine points adrift of the top six with nine games to play.
Davis said: "Never write this lot off - we could still make the play-offs.
"We haven't been consistent enough at times this season, but if we keep playing to the level we have in the last two games then who knows what can happen?
"The pitch was testing, but there was some good quality. We rode our luck when they hit the bar, but we dug in and it was a hard-earned win.
"We are showing grit and determination now and a great belief that we're never beaten and while we have the belief we always have a chance.
"Pogba was frustrating in the first half, but he got us back in the game and took his second well and I'm pleased for him."