In a tame opening half, Charlton looked the most likely to carve out an opportunity as George Tucudean looked for Igor Vetokele beyond the defence, but he was ruled offside.
Athletic went close just before the break as Vetokele saw his right-footed shot towards the top corner saved, having been picked out on the left side of the box by Yoni Buyens.
However, this did not deter the hosts and they took the lead early in the second half when Tommy Smith picked out striker Nahki Wells, who tucked in with his right from close range to make it 1-0.
The home side took charge of the second half despite having one man less and they almost cemented all three points late on, with James Vaughan being denied with a powerful shot heading towards the top corner.
His miss was to prove vital as Vetokele was able to rescue a point for the Addicks with the game's last attack, striking home from Andre Bikey's headed pass deep into injury time.