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Grabban almost put Bournemouth ahead with a right-footed shot from distance, but it just managed to evade the top left corner of the goal.
The 26-year-old forward was involved again soon after when he slipped in Matt Ritchie, whose left-footed shot was saved in the centre of goal.
Bournemouth dominated the opening half and it was Grabban who wasteful this time, as the striker fired wide having been picked out in the centre of the box by Andrew Surman.
The visitors took the lead shortly after half time. Matthew Gilks brought down Grabban in the box, leaving the striker to step up and slot home from the spot.
Bournemouth looked to put the game to bed with Marc Pugh and Ritchie both taking shots from outside the box, but neither was enough to really test Gilks.
The home side were almost able to steal a point late on as Kirk Broadfoot headed towards goal from a difficult angle, with the effort kept out by an impressive block as it made its way to goal.