Pitman's first hat-trick since August 2010 enabled Bournemouth to complete a comfortable victory for the home side, although Davis felt the referee had played a crucial part.
Teenager Ryan Colclough came on in the 63rd minute for Crewe and scored his first senior goal 13 minutes later to make the score 2-1, but Pitman's second penalty prevented any late revival.
Adam Dugdale was the first Crewe culprit, bringing down Marc Pugh to concede the spot kick, which Pitman gratefully despatched.
Bournemouth's second came in 67th minute when Pitman tapped-in a centre from Josh McQuoid.
But Crewe struck back with Colclough's goal from two yards after Matthais Pogba had shot across the face of the goal.
Pitman got his third in the 82nd minute when Abdul Osman was adjudged to have fouled Wes Fogden.
"We definitely feel hard done-by," said Davis. "The referee was their best player throughout. He changed the game in the first few minutes by giving a penalty and did it again in the second half.
"Bournemouth had the better of the second half, but that was only because of the goals they scored. We were the better team in the first half and kept going.
"It is difficult when those decisions go against you for a young side to keep going, but I thought we did well in those circumstances."
- Sports & Recreation