The side's Championship clash with Portsmouth at The Valley was paused in the 12th minute while a fan received treatment, before the game was subsequently postponed.
In a statement, the club said it was "devastated" at the fan's passing and thanked first responders and club staff "who did everything they could" to save the supporter's life.
"The club are grateful to supporters in attendance and the Portsmouth players and staff for their patience and understanding as this tragedy unfolded.
"Everyone at the club sends their heartfelt condolences to the supporter’s family and friends at this incredibly difficult time."
Portsmouth also paid tribute, with a statment on X saying the club are "devastated to learn that the Charlton supporter taken ill at today’s game has passed away.
"Our thoughts and deepest condolences are with their family and friends at this difficult time."