Daggers dominated the first half and were close to edging in front in the 16th minute.
Jon Nurse raced away from defender Inigo Calderon and his low drive could only be parried by goalkeeper Casper Ankergren into the path of Bas Savage, whose tame effort was miscued off the line by Marcos Painter, back onto Savage and up onto the crossbar.
The visiting keeper was called into action again several times before the interval, with Ankergren denying Savage, Nurse and Danny Green.
The hosts continued to look more threatening at the start of the second half, with Damian Scannell and Savage both going close, but it was Brighton who broke the deadlock in the 59th minute. Craig Noone crossed for Murray to volley in from six yards.
The visitors continued to hang onto their frail lead as Peter Gain had two efforts blocked, but Brighton held firm to move 13 points clear at the top.