The Daggers were coasting to victory at 2-0 at half-time, thanks to goals from Lee Goodwin and Chris Moore, but Mark Cooper's men showed tremendous spirit to hit back through Tris Whitman and Julian Alsop.
Tamworth could have got off to a great start after just three minutes when Whitman latched on to an Alsop flick-on and squared the ball for Bob Taylor who had a simple tap-in from close range, but somehow Scott Griffiths produced a goal-saving tackle.
The Lambs had not conceded a goal on home soil for nearly 12 hours of action, but fell behind in the 21st minute when Glen Southam's corner was turned in at the back post by Goodwin.
The Lambs' defensive frailties at corners were almost exploited again five minutes later, but Ashley Vickers' header flew just over the bar.
Dagenham doubled their advantage just before half-time when Goodwin slid the ball in behind the Lambs defence and Moore fired into the top corner from 12 yards out.
Tamworth needed an early goal at the start of the second half and Whitman obliged with a superb strike into the bottom right-hand corner of the net from 20 yards.
Then, with just four minutes remaining of normal time, The Daggers defence failed to clear a Simon Hollis cross and Alsop rolled the ball home past Tony Roberts from inside the six-yard box.