After a low-key first half, the home side ran riot after the break, scoring four without reply, although the visitors' cause was not helped with the sending-off of defender Solomon Tiawo seven minutes after the break, at a point when the game was still goalless.
But from that moment Shrewsbury grabbed complete control of the game as Ben Davies (twice), Marc Pugh and Marc Tierney were on target as the Shrews registered their third straight home win without conceding a goal to remain in touch with the League Two play-off pack.
Town in fact almost snatched a lead within the first minute, midfielder Stewart Drummond seeing his header tipped away at full-stretch by Daggers keeper Tony Roberts.
Throughout the first 20 minutes Shrewsbury pressed, but were kept at bay by either Roberts or some poor finishing, but they were unable to build on their early momentum and the half ended a scrappy, untidy affair.
But when Tiawo clattered into Davies in the 52nd minute, the visitors were to pay a high price. He was shown a red card, while Davies got to his feet to curl home the ensuing 20-yard free-kick.
In the 69th minute Davies put the issue beyond any doubt when he lashed home a smart, angled shot from 15 yards, with substitute Pugh scoring within a minute of entering the field, netting Shrewsbury's third with a close-range header on 77 minutes from a Tierney centre.
For lowly Dagenham things then became even worse as Pugh returned the compliment, crossing to the far post where Tierney was on hand to complete the scoring by heading home from six yards out.