Brett Pitman put the Cherries ahead from the penalty spot after a dubious decision from referee Richard Clarke before Mathias Pogba levelled for Alex.
But Pitman netted the winner six minutes from time to secure victory which saw Eddie Howe's side leap to the top of npower League One.
Crewe boss Davis said: "I was delighted with the all-round effort and performance from our players. It was an all-round great performance but we didn't get anything from it unfortunately.
"I have told the players it has gone and to get their heads up. As long as we keep playing like that and we get better decisions from the referee we will win more games than we will lose.
"Their keeper made some good saves and he'd have been my man of the match just behind the referee.
"Our lads were magnificent. They stood up to them, really opened them up and couldn't do any more than they did.
"We don't score many headers from crosses but we did here and that's good.
"There are plenty of positives to take apart from the result and we will take that into next Wednesday (for the club's Johnstone's Paint Trophy area final second leg against Coventry)."