They scored in the opening minute but had to fight for every ball in the following 89 in order to secure the vital three points that keeps their promotion hopes on track.
The only goal of the game came after only 29 seconds when recent loan signing Kevin Gall picked up possession on the edge of the area and lobbed goalkeeper Ian Dunbavin.
His shot bounced against the crossbar and the loose ball was despatched from close range by top-scorer Tommy Wright for his 12th goal of the season.
The early goal should have been a springboard to go on and win the game with ease, but it wasn't to be for Dave Penney's side.
The visitors almost grabbed an equaliser in the 14th minute when a 25-yard curling drive from Mark King was tipped over the crossbar superbly by Darlington keeper David Stockdale.
The home side continued in their quest for a goal that would have killed off the game and, in the 33rd minute, Wright found Julian Joachim on the right flank and his low cross was met by Michael Cummins, who fired narrowly over the bar from 12 yards.
Cummins went close again two minutes later but Dunbavin was equal to the task as he turned the midfielder's shot around the post.
In the second half the visitors went close through striker Ian Craney in the 47th minute and the home side came back on the hour when Joachim drove the ball across Dunbavin's line but his shot went the wrong side of the post.
The game was still on a knife-edge and the home side went close five minutes from time when a Clark Keltie corner was half cleared to Joachim and his shot was deflected wide for the corner.
From the resultant kick, defender Ian Miller headed narrowly over the bar.
Two minutes from time, Accrington almost grabbed an equaliser as Craney saw his long-range shot curl wide.