However, the hosts could have claimed all three points had striker Conor McAleny scored in just the first minute when he was put through one on one with visiting goalkeeper Andy Lonergan, or had Tony Watt kept his composure in the final minute of injury time.
But, moments after that save, Pope was at fault as the away side broke the deadlock in the 32nd minute. The young keeper failed to hold Ross McCormack's free kick and Ryan Tunnicliffe was on hand to turn in the rebound from close range.
El-Hadji Ba and Watt both had early second-half opportunities to draw Charlton level, but their failure to do so was punished in the 58th minute when McCormack kept up his positive goalscoring record against the Addicks.
On this occasion Tunnicliffe was the provider as the former Manchester United trainee picked out McCormack, who sidestepped one challenge before he fired a low shot into the bottom corner of Pope's net, taking his tally to eight goals in as many games versus the South Londoners.
As the game drew to a close Charlton manager Guy Luzon turned to Johnnie Jackson from the bench and with his very first touch the midfielder halved the deficit in the 81st minute. Johann Berg Gudmundsson's corner found the unmarked 33-year-old and he made the most of the space to head in.
It was a goal that sparked a number of half chances at both ends of the field, but it seemed that Fulham would hold on until the sixth minute of stoppage time when Cousins stooped at the back post to power in a header to salvage a share of the spoils for his side.