In-form striker Matt Blinkhorn grabbed his sixth goal in seven games with an injury-time winner for Morecambe.
Blinkhorn headed in Wayne Curtis's cross two minutes into added time to deny Dagenham a share of the points.
Jonathan Boardman's foul on Blinkhorn gave Morecambe a penalty on 70 minutes, but debutant Daggers keeper Ed Thompson saved from Carl Baker's effort.
Dagenham went close through Ryan Hall's 16th-minute header, while Shane Huke missed from close range on the hour.
Morecambe: Jalal, Yates, Bentley, Grand, Adams, Thompson, Drummond, Stanley, Baker (Curtis 76), Blinkhorn, Twiss (Newby 76).
Subs Not Used: Scott Davies, Howard, Hunter.
Goals: Blinkhorn 90.
Dag & Red: Thompson, Griffiths (Patterson 46), Uddin, Boardman, Foster , Hall (Sloma 63), Southam, Huke, Saunders (Taylor 89), Nurse, Strevens.
Subs Not Used: Hogan, Taiwo.
Ref: Karl Evans (Greater Manchester).