Abu Ogogo's equaliser ensured the points were shared in east London after Chris Zebroski put the visitors ahead with a superb finish from inside the area.
Zebroski's 25th-minute goal came after splendid work by Christian Jolley and Adam Chapman and the Welsh side went into half-time firmly in control.
Justin Edinburgh's side grew in confidence after a poor start and perhaps could have been more than a goal to the good at the break. Robbie Willmott volleyed over and Jolley was denied by a terrific save from Chris Lewington.
But the Daggers were a side transformed in the second period, it was almost a total role-reversal. Newport missed several chances, Lee Minshull the worst culprit, but the hosts were the dominant side.
And captain Ogogo swept home with 20 minutes remaining after a fine move started by debutant Zavon Hines.
They saw chance after chance go begging, defender Andrew Hughes cleared off the line magnificently at the death to deny substitute Afolabi Obafemi.