The home side should have taken the lead in the tenth minute when Dutch striker Clyde Wijnhard was fouled by Greg Strong in the area. The referee pointed to the spot but Wijnhard's spot-kick was saved by keeper Steve Wilson.
The visitors went close three minutes later when Paul Harsley's free-kick was headed over the bar by Michael Welch and then soon afterwards a long-range drive from Mike Sheron sailed wide.
The home side dug deep and they got their reward in the 25th minute when defender Matthew Clarke found Craig Hignett wide on the left. Hignett cut inside and picked out the on-rushing Wijnhard who fired low beyond Wilson from eight yards out.
In the 31st minute another free-kick from Harsley found Sheron, but again the effort was off target.
Darlington increased their lead five minutes from the break when captain Neil Maddison superbly volleyed a Hignett corner in from the edge of the area. But the visitors fought back and they got themselves into the game a minute later.
The home defence failed to clear a long ball from the right and Jonathan Parkin lashed the ball in from close range to make it 2-1.
In the second half Darlington's American midfielder Adolfo Gregorio came to his side's rescue with a 56th minute goal-line clearance to deny Sheron.
In the 70th minute Darlington should have had another penalty when Wijnhard was brought down inside the area. Amazingly the referee waved play on despite the protests of the Darlington players.
Wijnhard headed Maddison's free-kick narrowly wide in the 72nd minute, but the home side eventually sealed victory ten minutes from time.