The goal was scored by Chris Senior in the 119th minute, seconds before the match was due to end in a penalty shoot-out.
In the 18th minute, a Mansfield corner was met by the head of former Darlington captain Steve Foster, but the ball sailed well over the bar.
Despite Darlington dominating the opening exchanges of the second half, Mansfield created an excellent chance when Briscoe took on two players from the right and his cross-shot was deflected just over the bar by his team-mate Paul Connor.
[18] Both sides traded attacks, none of which came to much, before Darlington were awarded an 89th minute free-kick 25 yards out, following an unfair challenge on Verma.
[17][18] In the dying seconds of normal time, Bridge-Wilkinson crossed into the box where Wright headed wide with a diving attempt.
During a fraught final period of extra time Mansfield threatened when Ashley Cain's cross reached the far post where Connor headed narrowly over the bar.
In the 119th minute, an Aaron Brown long throw was flicked on by Miller and found Wright, who got to the ball ahead of Marriott to send a looping header towards the Mansfield goal.