After a quiet opening, the game sprang to life after 22 minutes when visiting keeper Tony Roberts allowed Kirk Hudson's low shot to beat him at the near post.
Aldershot extended their lead shortly after the interval, substitute Daniel Hylton chesting down Anthony Straker's cross into the path of Scott Donnelly, who fizzed a low drive past Roberts.
Dagenham then threw themselves a lifeline after 65 minutes when Paul Benson tapped home a Danny Green free-kick that had deceived home keeper Stephen Henderson.
And it was Benson again minutes later who levelled matters, twisting away from Chris Blackburn before drilling low past Henderson for the equaliser.
But the Daggers were not done there as Scott completed the remarkable turnaround late on.