Both sides had chances to score early on with Paul Bruce going close for the home side while Daniel Davidson forced Tony Roberts to make an early save.
John Still's men took the lead after 14 minutes. A long throw from Scott Griffiths ended up at the feet of Moore in the area. The striker turned immediately and fired a left-footed shot that found the back of the net.
The goal galvanised the Essex side as they produced an energetic and vibrant first-half performance, which only lacked a second goal.
Glen Southam came close to finding a second - his header from barely six yards clipped the post and went wide. The midfielder perhaps should have been looking to hit the target at the very least.
Tresor Kandol, on his return to Dagenham after a loan spell with Darlington, also went close while Bruce forced Scott Bevan into making another save.
The Daggers were made to pay for their missed chances as Tamworth forced an equaliser during the closing stages of the opening period. The dangerous Davidson clipped over a cross from the right that found an unmarked Matthew Williams, who duly headed home from eight yards.
The game could have gone either way in the second half. Davidson caused problems in the Daggers defence especially from set-pieces, and he saw a shot from a Scott Stamps corner cleared off the line by Griffiths.
The deciding goal arrived on 63 minutes. It certainly wasn't the prettiest of goals. The ball reached Moore from a Sam Saunders corner, and the striker scuffed the ball in from just about a yard from the goal.
Tamworth's best opportunity for an equaliser came from Williams who headed the ball against the crossbar from close range.
The visitors' frustrations were visible in the final minutes and they were reduced to ten men when Dave Bampton saw red after a clash with Dagenham midfielder Bruce.