Nabil Bentaleb was sent off just before half-time after picking up a second yellow card but the lads still managed to level at 2-2 before half-time.
They then struck back again for 3-3 after United took a 3-2 lead early in the second half only for the visitors to snatch the win five minutes from time.
Louis Rowley opened the scoring for United from a corner in the eighth minute.
We levelled after 21 minutes when Mason Bush and Josh Onomah combined to release Shaq Coulthirst down the right and his cross was deflected past United goalkeeper Pierluigi Gollini.
Both teams then struck the woodwork, Ben Pearson clipped the crossbar from a free-kick for United before Roman Michael-Percil curled a shot against the post.
It was soon 2-1 to United as Mats Daehli struck from close range and they had another chance as James Wilson skipped around goalkeeper Luke McGee only for Bush to touch his effort onto the post and Daehli then blast over the empty net.
Bush was soon up the other end setting-up a chance for Darren McQueen, who was denied by Gollini.
Bentaleb was then carded twice in 10 minutes, leaving the lads trailing 2-1 with 10 men.
Undetered, they hit back immediately as after a patient build-up, Michael-Percil crossed for McQueen to bury high past Gollini.
United were soon ahead in the second half as Wilson slid a pass into Paddy McNair.
We hit back for the third time in the 74th minute when Laste Dombaxe saw his shot blocked and after a scramble, the ball crept in off the post via defender Rowley.
United struck the winning blow with five minutes remaining as Pearson fired home from the edge of box.
There was still time for a final throw of the dice and Coulthirst agonisingly hit the woodwork in the final minutes.
"Even though we conceded two goals from set-pieces I thought we looked to be in a fair amount of control," said coach Alex Inglethorpe.
"Obviously the sending off has hurt us because it's more difficult trying to play with 10 men.
"That said, I thought we were magnificent at times and controlled large parts of the game.
"We equalised, went 3-2 down and to come back again with 10 men at 3-3 was testament to the players. I thought at the end we were unlucky not to make it 4-4.
"I thoughht Josh and Shaq were the stand-out players on the pitch, they were outstanding.
"It was a good performance and playing with 10 men was a good exercise for the lads."
Spurs U18s: McGee; Smith, Gardiner, Ball, Bush; Bentaleb, Onomah; Lameiras (Dombaxe, 54), Michael-Percil (Zapata-Caicedo, 46), D McQueen (Ogilvie, 75); Coulthirst.