Michael Morrison had the first chance of the game for Charlton, heading a corner narrowly over the crossbar.
It was indeed Charlton who opened the scoring, Fuller finishing from close range in the 15th minute.
After the goal Palace started to come forward, but were finding it hard to break down a dogged Charlton defence.
Charlton managed to hold on until the break to go in at 1-0.
They started to dominate after the break, rather than Palace get back in it, and Fuller could have added to his tally by testing Julian Speroni from 25 yards.
Instead, Palace got themselves back on level terms with Murray hammering into the roof of the net in the 74th minute.
Five minutes later the turnaround was complete and it was Murray again, this time firing in from close range.