Pitman was left unmarked to latch on to Liam Feeney's long throw in the 14th minute and fire past goalkeeper Russell Hoult on his Darlington debut.
And Pitman netted his fifth of the season from the penalty spot with eight minutes remaining after Darlington defender Tony Kane handled.
But Darlington were out of luck. Mark Convery was denied by a fine save from Bournemouth keeper Shwan Jalal two minutes after Pitman's opener.
Then on the half hour young winger Jamie Devitt brought another good save out of the keeper with a fierce drive towards the top corner.
The Quakers brought on veteran player coach Dean Windass who almost bundled the ball in on 65 minutes after good work from Devitt before Convery was denied by Jalal again. But Darlington were caught on the break by Anton Robinson and Kane had to handle to stop him.