The Daggers had Luke Howell sent off just before the break and they were made to pay as John Marsden and Danny Hattersley struck to secure a second-round tie at home to Eastleigh next month.
Dagenham striker Rhys Murphy hit the side-netting with a diving header in the fifth minute after getting on the end of a Damian Batt cross.
Southport goalkeeper David Raya Martin then dived low to his right on the half hour to push away Jamie Cureton’s goal-bound shot.
But in the 43rd minute Howell was shown a red card for pushing Luke George in the face after taking exception to a tackle by the Southport player in the centre circle, and the game swung in the Sandgrounders’ favour.
Southport took the lead in the 50th minute when Paul Rutherford cut the ball back to Marsden whose shot found the net with the help of a deflection.
Hattersley then doubled the hosts’ advantage from the penalty spot in the 58th minute, sending keeper Mark Cousins the wrong way after Matt Partridge had tripped Marsden just inside the penalty area.