Ross Turnbull, in the Boro goal, made a string of saves to deny the Spurs forwards throughout the first half.
But James Morrison hit the winner from 25 yards after a strong run, the ball beating the Spurs keeper after taking a wicked deflection.
Spurs had a goal disallowed in the dying minutes for handball.
The Boro Youth's progression has included wins over Bury, Barnsley and Luton. Now the reward for beating Spurs is a semi-final against Manchester City, who beat Mark Proctor's team 1-0 early in the season at Rockliffe.