Daggers started the game just two points above Town in the promotion zone and looked nervous from the off, with midfielder Solomon Taiwo epitomising his side's malaise with several misplaced passes.
Daggers brought the record home league crowd to their feet in the fifth minute when little midfielder Sam Saunders rose highest in the Shrewsbury box to head inches over from a Matt Ritchie cross.
Town nearly caught the home side with a sucker punch in the 16th minute when Chris Humphrey turned full back Scott Griffiths and sent in an inviting cross which Grant Holt powered straight at keeper Tony Roberts from point-blank range.
Holt made no mistake just two minutes later when he nodded another excellent Humphrey cross past Roberts from four yards to break the deadlock.
The impressive Humphrey turned goalscorer in the 32nd minute when he shook off Griffiths again on the edge of the penalty box before lobbing cleverly over Roberts and into the net to double his side's lead.
Daggers looked shaken but should have pulled one back in the 43rd minute when Saunders nodded narrowly wide from eight yards after latching on to a dangerous Ritchie corner.
The home side came flying out of the box in the second half, with Paul Benson coming close in the 50th minute when he thundered against a post from 10 yards.
Benson made no mistake in the 53rd minute when he out jumped defender Graham Coughlin to head a deep Ritchie cross past keeper Luke Daniels to pull one back for the home side.
Town soaked up the pressure and should have scored a third in the 68th minute when Holt scuffed wide from point-blank range after latching on to a clever cross from Humphrey.
Dagenham threw men forward in search of an equaliser late on, with Saunders firing agonisingly wide from eight yards in the 87th minute after weaving his way into the Town area, but Town held on to book a place in the play-offs.