Despite the home advantage, it was Bournemouth who came under most pressure in the opening exchanges, with Addicks forward Marvin Sordell coming within inches of marking his debut with a goal by heading over from a corner.
However, the Cherries would not have long to wait to see their first goal of the new campaign, with striker Grabban heading 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.
The Addicks were not level for long, though, as Grabban spun his marker on the edge of the area before unleashing an unstoppable shot past Hamer to grab his and his team's second of the afternoon.
Things went from bad to worse for Pritchard, who was dismissed moments later for a dangerous challenge on Bournemouth's Ryan Fraser.
The Addicks pushed for an equaliser despite their numerical disadvantage throughout the six minutes of stoppage time, but the hosts held on and marked their return to England's second tier with a win.