Pitman's first hat-trick since August 2010 enabled Bournemouth to complete a comfortable victory.
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.
Howe said: "We have played a lot of matches lately and with a heavy pitch, we looked leggy and tired.
"This was a big win for us because we had to grind it out. We were missing one or two players and others were carrying injuries. At times in the first half we looked dead on our feet.
"But such is the spirit and determination in the squad that we found our rhythm in the second half and were excellent.
"Brett Pitman's scoring record is good as anyone's. He worked hard and showed the right attitude after being left out of the side earlier this week."