The striker’s first touch after coming off the bench – a low spot-kick into the bottom corner – earned the Londoners a point after Danny Hollands had given the visitors a 64th-minute lead.
Grabban was given the penalty duties after Brentford missed two spot-kicks in their 4-2 win over Exeter at the weekend.
He struck three minutes after Hollands had given the Cherries the lead when he glanced home Rhoys Wiggins’ free-kick at the near post.
The visitors dominated in the early stages but it was Charlie MacDonald who went closest on the half hour when he drilled a low drive against the outside of the post.
Liam Feeney, Josh McQuoid and Wiggins wasted good chances to win the clash for Eddie Howe’s team with some wayward finishing.