Charlton controlled the first half and were unfortunate not to take the lead in the second minute when Lee Martin's shot hit the bar.
Simon Francis then had a header cleared off the line before the Addicks took the lead on 10 minutes as Miguel Llera headed home Johnnie Jackson's corner.
But despite Charlton's complete dominance, they paid the price for failing to kill the game off in first-half stoppage time when Therry Racon tugged back Bas Savage and the Daggers were awarded a penalty which Danny Green converted. The decision infuriated Charlton boss Phil Parkinson so much he was sent off and forced to watch the remainder of the game from the stands.
Only a stunning one-handed save by Ross Worner prevented Green's header putting Dagenham in front but Charlton regained the lead in the 90th minute when Johnnie Jackson headed home Racon's cross.
However, there was still time for Nurse to connect with Damian Scannell's cross as Dagenham escaped with an unlikely draw.