Callum Howe's 55th-minute goal looked like it would be enough to earn Simon Weaver's team back-to-back 1-0 victories, until Conor Wilkinson popped up at the death to level matters.
An encouraging first-half performance saw the hosts dominate possession and pass the ball as slickly as they have in some time.
A Josh Falkingham strike and Mark Beck's glancing header brought routine saves out of visiting stopper Elliot Justham with Town making all of the early running.
Beck then stretched out a long leg to stab Jack Emmett's pull-back from the right byline towards goal, though his effort was blocked inside the six-yard box.
Harrogate got plenty of joy raiding down the right flank, and, from one such move, the ball broke for Jack Muldoon on the edge of the box where he unleashed a snap-shot that beat Justham at his near post but thudded to safety off the foot of the upright.
The opening period ended with Town really pressing and Emmett whipped one just millimetres over the Daggers' cross-bar.
The second 45 began in exactly the same fashion, Weaver's men enjoying almost exclusive possession of the football as the East London outfit dropped progressively deeper.
Warren Burrell saw a goalbound effort hacked off the line by a visiting defender after play was recycled following a right-wing corner.
The breakthrough then arrived with 55 minutes on the clock.
The ball looped up inside the box for Muldoon to force a parry out of Justham with a low shot and Howe was on hand to confidently slot the rebound into the bottom corner of the unguarded goal.
Muldoon was causing further problems for the Daggers moments later, using his pace to accelerate onto a ball in behind, then showing his strength as he cut inside from the left and bent a strike over the top.
An excellent 75th-minute burst by Emmett then left Muldoon with just the keeper to beat on the right edge of the area, however Town's top-scorer got his finish all wrong.
The hosts would eventually pay for not capitalising on their dominance as Dagenham eventually showed some attacking intent in the final moments.
Substitute Olly Harfield forced James Belshaw into a decent stop with a 90th-minute drive at goal before disaster struck two minutes into added-time.
A low Harfield cross from the left found its way through to Wilkinson, who had too much time inside the six-yard box to pick out the top corner.
Three points would have lifted Town up to sixth place in the National League, however the visitors' late strike sees them drop one position to eighth and out of the play-off spots.