The visitors began the match in a positive vein and threatened on a couple of occasions early on, Lewis Grabban firing over from a corner and Harry Arter forcing a save from Ben Hamer with a swerving strike.
Charlton soon settled into the game, however, and created a chance of their own in the 25th minute as Lee Camp spilled Danny Green's effort before recovering just in time to beat Simon Church to the rebound.
The hosts came even closer to taking the lead five minutes before the break as Johnnie Jackson exchanged passes with Green before almost catching the keeper out with an effort that came back off the post.
They threatened again in the 44th minute when substitute Jon Obika fired a strike towards goal on the turn that Camp needed to save.
Bournemouth almost broke the deadlock just after the hour mark when Marc Pugh found space to shoot from the edge of the box, but his curling effort was too high to trouble the keeper.
The visitors had another chance less than 10 minutes later when they were awarded a free kick on the edge of the box. Ian Harte stepped up to take it, only to fire an effort over the bar.
Charlton came close themselves shortly afterwards when Obika showed good strength to tee up Jordan Cousins, who proceeded to blaze his shot over the top.
The Cherries threatened once more just two minutes later as Tommy Elphick rose highest in the area to meet a corner, but the centre-back's header went wide.
The Addicks responded with a chance of their own in the 80th minute, but Jackson and Church could not beat Camp, who kept the duo out with some inspired goalkeeping.
Jackson came close again with time running out before the hosts finally got the winner in stoppage time as Dorian Dervite gave his side three massive points.