Ross Turnbull struck in the third minute as the Ammies held on to reach the second round for the second consecutive season.
After the match, the second round draw for the competition was made, with Salford being drawn at home against Chesterfield. The ties will be played across the weekend of Friday 3 until Monday 6 December. You can see the FA Cup second round draw in full here.
Story of the match
Salford grabbed the only goal of the match when Turnbull headed home a corner past Elliot Justham.
Dagenham & Redbridge went searching for an equalizer as Angelo Balanta played Mauro Vilhete, but he was deemed to have been just offsides.
Mo Sagaf had a go from 20 yards out that was off-target and subsequently had a penalty appeal waved away. After a series of corners, Matt Robinson's shot flew over the bar.
Salford nearly doubled their lead after a poor pass by Myles Weston saw Aramide Oteh in, but a brilliant challenge by Kenny Clark kept him out.
The hosts should have been level when Robinson got on the end of a Sam Ling cross, his header brilliantly saved by Connor Ripley. He then denied Weston and Robinson just before halftime.
Oteh sliced his effort just wide after the break and at the other end, Sagaf was again off the mark from distance.
Ripley spared the Ammies blushes when he kept out a Callum Reynolds header from a Will Wright delivery. He then kept out George Saunders' shot with his fingertips.
The Daggers had one final chance to level and force a replay when Wright's free-kick trickled to the back post and found Robinson, but he went over and Salford were through.
Man of the match: Connor Ripley
The Salford goalkeeper, on-loan from Preston North End, came up with some incredible saves precisely when his side needed it, especially in the second half when Dagenham & Redbridge turned up the pressure.