Six minutes before the end an ugly melee developed as a result of an altercation between Chris Moore of Dagenham and Tamworth's Sekani Simpson.
David Bampton, who was already on a yellow card, received his marching orders for seemingly throwing a punch during the flare-up. He was followed to the dressing room in stoppage time by Daggers skipper Ashley Vickers for a second bookabke offence.
The first half was a largely tame affair, with chances few and far between.
In very testing conditions, both defences remained concentrated as the half ended goalless
Craig Mackail-Smith had an excellent opportunity to open the scoring in the 15th minute. Ashley Vickers headed the ball to Mackail-Smith, but the Dagenham striker shot wide.
Tamworth were almost punished in the 22nd minute when Adie Smith was caught in possession, but the Daggers were unable to capitalise on his mistake.
The Lambs showed some attacking flair of their own, with Louie Soares and Tristram Whitman looking lively, but they were lacking the final pass to trouble Tony Roberts' goal.
Ten minutes before the break, Dagenham were denied by an excellent save from Lambs keeper Ryan Price as Lee Goodwin fashioned a good chance for Mackail-Smith on the edge of the area.
Midway through the second half Soares twice tested the Daggers defence but managed to blast over the bar on both occasions. In injury time, Glen Southam almost stole it for the home side, but Price once again saved well low down.