Daggers dominated the first half and made a dream start when Carlisle defender Lubomir Michalik conceded a penalty for a foul on John Akinde in the fourth minute.
Danny Green's penalty was brilliantly saved by goalkeeper Adam Collin diving low to his left, but Nurse fired in the rebound.
Daggers doubled their lead in the 29th minute when Akinde's low shot was parried by Collin into the path of Nurse, who tapped into an empty net from two yards.
As Carlisle searched for a way back in the second half, their best chance fell to substitute Paddy Madden. He outmuscled defender Femi Ilesanmi, but his low drive was pushed away by goalkeeper Tony Roberts.
John Still's side sealed victory in the 84th minute Akinde squared to Green, who powered his effort low into the bottom right corner before he was dismissed moments later following a second bookable offence.