The visitors began the afternoon the brighter of the two teams, with Addicks striker Marvin Sordell coming within inches of marking his debut with a goal some five minutes after kickoff by heading over from a corner.
It was not until the 10th minute when the first real scoring opportunity presented itself for the hosts, with debutant Mohamad Coulibaly shooting narrowly wide from the edge of the area.
However, the Cherries would not have long to wait to score their first goal of the new campaign, as striker Grabban headed home Harry Arter's cross on 26 minutes.
Arter, who enjoyed a two-year spell with the Addicks, came close to piling further misery on his former club four minutes later, but his long-range effort was well held by Charlton goalkeeper Ben Hamer.
Chris Powell's men came within a metre of restoring parity, with Charlton winger Callum Harriot cutting inside before unleashing a low shot from the edge of the area, but Cherries keeper Ryan Allsop watched the effort fly wide.
Grabban went close to scoring his and his team's second goal seven minutes from the interval, but his swerving shot was dealt with by Hamer.