The Cherries had led 2-0 at half-time through Marc Pugh and Anton Robinson, who both scored in the opening 20 minutes.
Bournemouth played crisp, attractive football throughout and took a 10th-minute lead when Liam Feeney picked out Pugh to score with a header at the far post. Ten minutes later Michael Symes combined with Robinson, who doubled Bournemouth's lead.
The first of Pitman's hat-trick came four minutes after the break with Pugh the provider.
Bournemouth's Marvin Bartley made two amazing goal-line clearances before Pitman pounced on a defensive blunder by Peterborough substitute Charlie Lee to make it 4-0.
Dozens of traveling Posh fans headed for the exits before Kelvin Langmead scrambled a consolation reply after an hour and five minutes later Pitman completed his hat-trick with a blistering fifth goal for Bournemouth.