Grabban, 25, found the back of the net either side of a stunning Yann Kermorgant scissor-kick to give the Cherries all three points on their return to the second tier of English football.
It was the London side who started brightly and left-winger Callum Harriott fired the game's first chance over the cross bar after six minutes.
But Bournemouth responded well and after new French recruit Mohamed Coulibaly had curled a right-footed effort wide, they took a deserved lead after 26 minutes.
The former Charlton academy prospect curled in a left-foot cross which Grabban rose highest to meet and guide past a stranded Ben Hamer.
Former Charlton star Scott Parker would have been proud of his brother-in-law Arter, who was the driving force in Bournemouth's midfield.
The promoted side were flying and Wes Thomas was inches away from doubling the lead minutes later but his sliding effort from a Pugh cross whistled just wide.
Grabban then went on a duel with his nemesis Hamer but came off second best with the goalkeeper denying the forward's next three attempts.
Charlton were looking to carry on from where they left off last season where a good run of form saw them finish just three points outside the play-off places.
As they pushed for an equaliser their new loan signing, former Great Britain Olympic striker Marvin Sordell, exchanged passes with Harriott, who dragged a left-foot drive wide of Ryan Allsop's post.
Straight after the break they were back on level terms. Shaun MacDonald deflected a Harriott cross into the path of Kermorgant, who acrobatically rifled a scissor-kick past a stunned Allsop.
But they were not level for long. Grabban, who scored 13 goals last season, collected the ball after 66 minutes and drove right-footed past an outstretched Hamer to send the home fans wild.
Wales international Simon Church, who only signed for The Valley club in the week, was introduced soon afterwards but failed to inspire an equaliser.
The closest they came to grabbing a point was when Pritchard had a header cleared off the line by Cherries new boy Elliott Ward after 80 minutes.
Moments later the result was all-but sealed when Pritchard took an early bath for his frustrated lunge on Fraser.
- Sports & Recreation
- Bradley Pritchard
- Callum Harriott
- Yann Kermorgant