Darlington almost went in front on eight minutes when Tom Robson's low drive brought a fine save out of Newcastle keeper Alex Baird.
Quakers had keeper Jordan Nixon to thank for a reflex save with his legs to keep out Marcus Maddison's close range shot on 20 minutes.
But Newcastle were in front on 29 minutes when Remie Streete headed in a corner after leaping high beyond the far post, though the ball took a deflection to beat Nixon.
Newcastle dominated possession and on 50 minutes Nixon made a superb save, turning Knight's long range shot against the bar.
On the hour Quakers were almost level. Jamie Barton raced down the left and crossed for Danny Lambert, whose shot was half saved by Baird, with a defender scrambling back to hack it away before it crossed the line.
on 72 minutes Newcastle doubled their lead, Maddison firing low into the corner from the edge of the penalty area.
The game was up on 78 minutes when Adam Campbell fired in a third from a free-kick.