Steve Thompson's injury-time strike secured victory for Burnley and left Charlton nine points from safety.
Bottom-placed Charlton had taken a first-half lead when Nick Bailey tucked away a cross from Tom Soares.
But with 14 minutes remaining Thompson equalised from close range after Martin Paterson's headed assist.
And in the 94th minute Kevin McDonald found space on the right flank and crossed in for Thompson to fire home from six yards out for his second goal.
Burnley manager Owen Coyle: "The courage and bravery the players showed was tremendous. There was real bravery in their performance. We have the courage of our convictions.
"It was important to win today, we have played 40 games and are just two points away from the play-offs.
"It would have been easy to hide at 1-0 down but my players don't know when they are beaten. We have great spirit at this football club."
Charlton boss Phil Parkinson: "To put in the performance we did and get nothing is a blow. Six minutes of stoppage time was a joke. The referee got sucked in by the home supporters - I expected four minutes and it's cost us.
"We came under pressure in the second half, but we had two moments where we didn't work hard enough to stop the ball coming into the box.
"Whilst we showed resilience and commitment, that has to go right down to the wire."
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