The hosts established the lead in the first five minutes through striker Ellis Lehane, before a double from Luca Williams-Barnett saw them head into the break with a three-goal advantage.
It looked like the game had slipped away from United when Williams-Barnett completed his hat-trick on the other side of half-time.
Substitute Shea Lacey's brace gave Adam Lawrence's side a fighting chance however, and Gabriele Biancheri's converted penalty in the final five minutes made for a shaky finale for Spurs, but time ran out before we were able to complete an unlikely late comeback.
The hosts made a rapid start to the tie through their striker, Lehane, who latched onto Williams-Barnett's through-ball and dispatched a strike beyond our goalkeeper William Murdock's grasp.
United grew into the game following the opener and produced our best chance of the half in the 15th minute. Jayce Fitzgerald delivered a corner to the head of Reece Munro who beat his marker, and the Spurs goalkeeper, but Lehane was on hand to sweep the effort off the line.
Following that near leveller, the north London side further extended their lead.
Williams-Barnett first doubled his side's lead, knocking in a rebound from Lehane's parried effort, before doubling his personal tally for the game a little over five minutes later, dribbling into our penalty area from the left before slotting home through a crowded six-yard box.
The young Reds emerged from the break with plenty to do after the hosts' three first-half strikes, but it was Spurs who notched first in the second period, again through Williams-Barnett to seal his hat-trick.
Substitute Shea Lacey entered the field on the hour mark and gave Tottenham plenty to worry about despite their comfortable lead. His first goal came in the 78th minute when he cut in from the right wing and squeezed an accurate finish into the goalkeeper's bottom-left corner.
Seconds later he had his name on the scoresheet again, firing a shot that proved powerful enough to slip through Sam Archer's hands and into the net. There may have been some arguments on whether the ball crossed the line, but Amir Ibragimov smashed it into the back of the goal to be on the safe side.
A penalty in the final five minutes, coolly placed beyond the goalkeeper by Gabriele Biancheri, gave the hosts a nervy finish with seven minutes of injury time added on, but our late flurry of goals would arrive just too late to complete the comeback.