Pitman was left out as he mulls over a move away from the club, but the Cherries ploughed on regardless as two first-half goals and a second just after the hour saw them home.
Josh McQuoid outpaced the home defence following a long clearance from Michael Symes to fire home pass advancing keeper Joe Collister after eight minutes to open the scoring.
The Cherries doubled their lead on 18 minutes – Symes netting easily from the penalty spot after he had been tripped by Ian Goodison.
Tranmere made heavy weather of things before Aaron Cresswell saw a 39th-minute free-kick from the edge of the box deflected wide. Then Enoch Showunmi headed off target and Joss Labadie had a shot saved by Shwan Jalal approaching the break.
It was all over for Tranmere on 61 minutes when the Cherries added a third in a quick break from their own half. Marc Pugh netted off the far post with an angled shot when set up by the impressive McQuoid.