Charlton took the lead after 22 minutes when Kyel Reid's low cross was fired past Cherries keeper Shwan Jalal by Sodje from six yards.
Reid twice went close to doubling their advantage, first with a long-range free-kick before then firing wide after breaking into area.
Bournemouth struggled to create any real opportunities and they had to wait until the 55th minute to really test Addicks keeper Rob Elliot when he was forced to gather Liam Feeney's low drive from 20 yards.
The Addicks were reduced to 10 men just after the hour mark when Jose Semedo was handed a straight red card for violent conduct after raising his studs in a challenge on substitute Michael Symes.
Bournemouth had a 76th-minute penalty claim ignored when Symes' shot was deflected behind by Gary Doherty for a corner, although the referee adjudged it had not come off the Charlton defender's arm despite furious protests from the visitors.