Aaron Williams handed Simon Weaver’s high-flyers a half-time advantage with his third goal in as many games, but an Inih Effiong brace put the visitors in charge before Langmead’s late, late intervention.
From a Town perspective, the opening 45 minutes were fairly flat both on the pitch and in the stands.
It was the visitors who played with the greater urgency and gigantic striker Effiong stung the palms of Harrogate custodian James Belshaw in the 12th minute when the ball broke to him on the edge of the home box.
The hosts eventually began to generate some forward momentum, visiting stopper Mitch Walker forced to make a double-save by Liam Kitching following Joe Leesley’s corner, George Thomson then seeing his follow-up blocked.
With 26 minutes on the clock, Weaver’s men managed to get their noses in front.
Leesley’s free-kick from the left was met at the back post by Callum Howe, who headed across goal where the lurking Williams buried a finish from just a couple of yards out.
Langmead then made a brave block to keep out Anthony Jeffrey’s goal-bound effort at the other end, before Leesley’s swerving 20-yarder was parried unconvincingly by Walker.
The second period began disastrously for Town.
Barely a minute had passed following the resumption when Jeffrey did Warren Burrell for pace down the Dover left and crossed for Effiong, who span and poked the ball past Belshaw.
The same player came within a whisker of grabbing a second moments later, but drilled narrowly wide of the target.
Home goal-scorer Williams had a strong shout for a penalty turned down when he was wiped out inside the six-yard box and Langmead looked to have moved Town back in front following a fine leap and header from a right-wing corner, only to see his effort nodded off the line.
Dover’s Jamie Allen then scuffed a strike into the ground and against the bar from Effiong’s looping cross before the latter turned the game on its head.
Weaving his way into the box from the left, he held off a number of would-be tacklers and managed to force a strike past Belshaw at his near post.
Captain Josh Falkingham saw a low effort from long range held by Walker and another Langmead header was superbly tipped over by the Athletic gloveman as Town looked for a way back into the game.
In stoppage-time, Thomson lashed over the top from close range with the goal gaping and Kitching glanced a header wide when well-placed as the hosts turned the screw.
However, with four minutes of the additional six played, it seemed as though Dover’s time-wasting tactics had paid off and they were going to hold out for just their second win of the campaign.
Langmead, dangerous in the away area throughout the second half, had other ideas however, guiding a header past Walker to level things up in dramatic fashion.
There was still time for Jeffrey to get himself sent off after he completely lost control and squared up to Thomson, his own team-mates struggling to hold him back before the referee called time on proceedings.
The result sees Town miss out on the chance to move top of the pile and drop one place to fourth in the National League standings.