Lambs would be rock bottom of the table if it was not for Crawley Town's points deduction for going into administration. They had conceded seven goals prior to this game.
Dagenham and Redbridge were looking to get back to winning ways after their narrow defeat to Conference new boys Oxford in midweek.
The Daggers started the game at a fierce pace and dominated the first half but it was the visiters who had the first chance. Tamworth new-boy Weaver played a long-ball to Jon Stevenson in the opening minute, he took the ball on the chest and volleyed from 20-yards but his shot was comfortably saved by Tony Roberts.
The home side took the lead in the 4th minute when captain Anwar Uddin's deft header from a Sam Saunders free-kick beat James Dormond at the far post for his second goal of the season.
Dagenham continued to press and in the 16th minute they added a second. Glen Southam's low cross from the right was missed by Mackail-Smith but the ball rolled across to the penalty spot and Sam Solma fired into an empty net.
Stevenson looked the only danger for the Lambs and his shot midway through the first half was well blocked by Uddin.
Tamworth had a rare chance to halve the deficit in stoppage time of the first half but Graeme Ward's curling shot from long-range was comfortably saved by Roberts.
The second half was much the same as the first as the home side continued to dominate.
Mackail-Smith made a probing run into the Tamworth half and crossed to Akurang on the edge of the six-yard box, his shot was turned away for a corner by the feet of Dormand.
The Daggers added the third from the resulting corner, Saunders in-swinging set-piece was headed in at the far post by Akurang.
The two players combined again on 67 minutes to add the fourth, a free-kick from the left-hand side by Saunders found the head of an unmarked Akurang who grabbed his second of the game.