Addicks boss Chris Powell made six changes to the side which ended Huddersfield's 43-game unbeaten league run on Monday, but his fringe players disappointed as Charlton created very little aside from a 38th-minute close-range shot from Danny Hollands, which was well saved by Adam Collin.
Carlisle looked dangerous going forward and Matty Robson almost fired them ahead within seconds of the restart before Collin was forced to tip over Bradley Pritchard's shot at the other end.
Both sides improved dramatically after the break, but it was Charlton who broke the deadlock in the 64th minute when Morrison prodded the ball home from close range after his initial effort had been blocked.
Carlisle twice went close to equalising as Rhoys Wiggins was forced to clear substitute Francois Zoko's effort off the goalline before Sullivan tipped over James Berrett's shot.
But it was Charlton who sealed victory in the 90th minute as Yann Kermorgant's cross picked out fellow substitute Euell at the back post and he converted from three yards.