Striker Paul Benson celebrated his return from injury by scoring in a 70th minute to cancel out a first-half Shrews opener.
The home side bossed the first half with Mark Arber volleying high and wide in the first minute, after latching on to a fluffed clearance on the edge of the Shrews box.
Midfielder Glen Southam tried his luck with an audacious volley in the tenth minute which flew inches wide of target.
Ben Strevens should have scored in the 15th minute when he headed weakly at Shrews keeper Glyn Garner, after latching on to a clever Jon Nurse cross.
The Daggers upped the pressure in search of a goal with full-back Scott Griffiths volleying wide from long range in the 26th minute.
Shrewsbury weathered the storm and took the lead against the run of play in the 34th minute, when Andy Cooke scrambled home from six yards after Daggers failed to clear a corner.
The home side thought they had equalised in the 40th minute only to see Strevens' close-range strike ruled out for offside.
Nurse came close to levelling the scores in the 55th minute when he thundered a bullet header narrowly over from a deep Griffiths cross.
Asa Hall nearly doubled the Shrews lead 12 minutes later when he blasted a clever volley wide of Tony Roberts' goal from 25 yards.
Dagenham equalised in spectacular fashion in the 70th minute when Benson latched on to a clever Strevens nod down to fire past Garner from six yards.
Peter Gain tried his luck from long range in the 74th minute when he shot straight at Garner from 30 yards.
The home side should have bagged all three points in the 88th minute when Southam fired weakly at Garner to finish off a clever attacking move.
Dagenham upped the pressure in search of a winner, but Town held on for a point.