Goalshy Dartford failed to score for the third successive league match on Tuesday night.
The National League South leaders, who lost their unbeaten record by the same scoreline at big-spending Dorking Wanderers last weekend, lacked confidence in their play against Concord as they were beaten for the second successive match.
After weathering an early Dartford storm, Concord defended well with five across the back and Dartford struggled to create many openings.
Josh Hill's downward header was not converted in the fifth minute before Dartford top scorer Ade Azeez beat Jayden Randell and cut in from the left but shot straight at keeper Myles Roberts.
Noor Husin had a couple of long-range attempts off target but that was the sum of Dartford's first-half chances despite plenty of possession.
Concord took their time to warm up on a chilly evening in Essex but they looked a threat on the break with Temi Babalola causing problems only to choose the wrong option too many times.
Dartford's best chance of the game came seven minutes into the second half. Danny Leonard played a one-two with Kieran Murtagh down the right and his cross picked out Dan Roberts, whose near-post effort was turned round the post by Roberts.
The only goal of the came arrived just past the hour. Lemar Reynolds got the wrong side of Jordan Wynter and then twisted skipper Tom Bonner inside out before firing low beyond keeper Dan Wilks into the far corner.
It could have been worse for Darts with 15 minutes left as Reynolds was played in by James Blanchfield but Wilks produced an excellent one-handed save to keep his side in the game.
Darts introduced Luke Wanadio, George Porter and Emmanuel Sonupe but all they could muster was a wayward Wanadio shot in stoppage time.
Concord: Roberts, Reid (Popo 81mins), Randell, Pollock, Kensdale, Payne, Blackman, Blanchfield (Peck 88mins), Green, Reynolds (Martin 90mins), Babalola. Subs not used: Gordon-Stearn, Roast.
Dartford: Wilks, Wynter, Hill, Bonner, Meade, Leonard (Wanadio 76mins), Murtagh, Allen (Sonupe 76mins), Husin, Azeez (Porter 81mins), Roberts. Subs not used: Jebb, Essam.
Referee: Steven Hughes.