The Addicks led twice, with Chelsea loanee Conor Gallagher putting the visitors into a deserved lead in the 41st minute, but Fulham improved in the second half and were level in the 55th minute when Ivan Cavaleiro fired home.
Charlton went back in front two minutes later through Macauley Bonne's header, but Fulham equalised once again when Mitrovic scored from close range.
It was Scott Parker's first game as a manager against the club where he started his career and the travelling Charlton fans were quick to let him know they still have not forgiven him for his big-money move to Chelsea in January 2004.
The first action of the game came in the ninth minute. A lucky bounce sent Bonne clear of the Fulham defence, but Marcus Bettinelli raced off his line to make a brave block at the feet of the striker.
Fulham had to wake up in this early Saturday kick-off and Bettinelli parried away a Jonny Williams drive just after the 30-minute mark before the visitors went in front. The ball was worked out wide to Jake Forster-Caskey on the left and his deep cross was nodded down by Chris Solly for Gallagher to sweep home from eight yards.
And the Cottagers were level when Cavaleiro played a one-two with Decordova-Reid down the right before cutting inside and hitting a delicious left-footed shot past Dillon Phillips.
But the parity was short-lived as Charlton went back in front two minutes later. The hosts failed to clear their lines from a corner, and when Darren Pratley eventually clipped the ball back in, Bonne rose to flick a header over a stranded Bettinelli.
Joe Bryan and Mitrovic both rattled the frame of the goal for Fulham but it was all square again soon after.
Decordova-Reid's cross was met by Tom Cairney, who saw his header from close range brilliantly saved by Phillips. Mitrovic was there, however, to gobble up the rebound, scooping his shot into the roof of the net.
The pulsating action continued and Bettinelli made a fine save to tip substitute Jonathan Leko's shot behind in the 73rd minute.
Phillips then made an excellent stop to get a touch on Decordova-Reid's curler and send it over, before Cairney dragged a half-volley just wide.
With seven minutes of stoppage time announced, Phillips made an incredible save to deny Decordova-Reid from point-blank range when the on-loan Cardiff man met Cavaleiro's cross as the game ended honours even.