The match went to extra-time as Boro more than held their own against the Premier League side.
Teenager Josh Coburn then came off the bench to score the winning goal in front of a sell-out crowd.
It means Boro have now beaten Manchester United and Spurs to reach the last eight of the competition.
Here’s how each Boro player fared at the Riverside Stadium:
A message from the Football Clubs Editor
Our aim is to provide you with the best, most up-to-date and most informative Middlesbrough coverage 365 days a year.
This depth of coverage costs, so to help us maintain the high-quality reporting that you are used to, please consider taking out a subscription to our new sports-only package here.
Your support is much appreciated. Richard Mennear, Football Clubs Editor
1. Joe Lumley - 8
Had a few nervy moments when he came off his line but made some crucial saves, not least an alert stop to keep out a Tottenham corner in extra-time. 8 Photo: Robbie Jay Barratt - AMA
2. Anfernee Dijksteel - 8
Did an excellent job marking Son Heung-Min. Managed to provide support when Boro were on the attack from the right of a back three. 8 Photo: Getty Images
3. Dael Fry - 8
Marshalled Harry Kane throughout and was dominant in the air. Another fine showing against a world-class striker. 8 Photo: Robbie Jay Barratt - AMA
4. Paddy McNair - 8
Comfortable stepping out with the ball and made some vital interceptions on the occasions Boro’s backline was stretched. 8 Photo: Robbie Jay Barratt - AMA