Substitute Martin Smith smashed in a superb 30-yard shot to send Darlington into the northern area semi-finals of the Johnstone's Paint Trophy.
Smith volleyed in the winner with 15 minutes left to make it nine games unbeaten for the Quakers.
Neil Wainwright went close for the home side in the first half but his shot came back off the post.
Simon Brown had the visitors' best chance but his shot was well saved by keeper Sam Russell.