The former Arsenal trainee sent a neat, chipped finish over the head of the advancing Shrimpers goalkeeper Daniel Bentley in the 31st minute of the match.
And that came just four minutes after Southend defender Mark Phillips was sent off for a reckless challenge on Murphy.
The Southend defender had not been dismissed in eight years but was shown a straight red card by referee James Linington following the late foul on the forward.
After the break, Murphy missed a good chance to double the Daggers' lead before Medy Elito volleyed inches wide of the target.
The closest the Shrimpers came to scoring was a powerful 20-yard shot from substitute Freddy Eastwood, which was pushed wide of the right post by Chris Lewington.
But Dagenham were forced to work hard for the win, which came just two weeks after they beat Southend 5-2 in the Johnstone's Paint Trophy at this venue.