Pitman put the Cherries ahead from the penalty spot at Gresty Road after a dubious decision from referee Richard Clarke before Mathias Pogba levelled for the Alex.
But Pitman netted the winner six minutes from time to secure a 2-1 win which saw the Cherries leapfrog Doncaster into top spot.
Bournemouth boss Howe said: "We showed real team spirit and togetherness.
"I think away from home it is easy to find ways to get beat but the best teams always find a way to win.
"We scored a late goal and away from home that's not always easy. Our fans were magnificent and they would have given the players a big boost at crucial times.
"Both sides had their moments but our keeper (Shwan Jalal) made some greats saves for us at key moments and we defended well.
"Crewe are a very good team as we saw a few weeks ago so we were going to play an extra man in midfield as both teams liked to keep possession."