The match was a dreary affair prior to Pitman's opener, which Alan Connell created from the right of midfield, picking Pitman out in space on the left. Bournemouth's top scorer took his opportunity superbly, sending a low shot across the bows of Casper Schmeichel.
But County, who had failed to fashion a chance from the first half, looked a far different proposition when manager Hans Backe switched formation to move Westcarr up alongside Hughes.
The pair struck twice in quick succession to see County through - Hughes equalising three minutes into the half and Westcarr scoring the winner, set up by substitute Matt Ritchie.
Bournemouth seldom looked like saving the tie as torrential rain totally waterlogged the surface in the second half.