They would have won the match in the last few minutes but for an incredible miss by striker Luke Varney.
After their midweek home defeat by Bristol City, Charlton boss Alan Pardew made five changes giving a debut to 18-year old midfielder Josh Wright and recalling striker Izale McLeod for his first game since March following a serious knee operation.
But the home side found themselves a goal down after just 13 minutes when Charlton failed to clear a corner and Joey Gudjonsson's shot from outside the area was touched into the net by Steve Thompson.
Burnley thought they had scored after only eight minutes when Martin Paterson netted but, after first giving the goal, the referee changed his mind after consulting a linesman.
Charlton were awful in the first half and failed to get a shot on the Burnley goal.
The nearest they came to scoring was from a Grant Basey free-kick which was blocked.
Pardew made two changes at half-time bringing on Lloyd Sam and Svetoslav Todorov and it immediately had an impact as the home team surged forward.
Striker Varney headed straight at Burnley goalkeeper Brian Jensen and the home side eventually got the equaliser on 76 minutes when Todorov home Basey's free-kick.
Burnley almost snatched a second when Gudjonsson's shot beat everybody but rebounded back off the crossbar.
After Darren Ambrose had fired just wide for the home side, Varney produced the miss of the season as he intercepted a Burnley defensive mistake, took the ball round goalkeeper Jensen and with the empty net gaping took so long to put his shot goalwards that the visitors were able to scramble the ball to safety.