Jon Nurse and Josh Scott grabbed the goals for John Still's men either side of half-time as they booked a two-legged clash with Morecambe.
Daggers top-scorer Paul Benson had a cross-shot tipped over by Darlington goalkeeper Nick Liversedge after the first quarter of an hour.
But Liversedge was then at fault for Dagenham's 32nd-minute opener as he failed to come for a Danny Green corner, allowing Nurse to head home from six yards.
Green almost doubled Dagenham's lead three minutes after the interval as his free-kick forced Liversedge into action.
Darlington's Andrew Milne was then denied by Tony Roberts before Dagenham made the points save in the 75th minute. Dan Burn was dispossessed by Benson and his low cross was side-footed in from close range by Scott.