Dominic Solanke’s double, either side of a Dejan Kulusevski goal, had seemingly given Ange Postecoglou’s side an unassailable 3-0 lead, with an hour gone.
But substitutes Joshua Zirkzee and Amad capitalised on two Fraser Forster mistakes to set up a grandstand finale, in what was now a nervy north London.
United came within a whisker of scoring an equaliser before Son Heung-min netted a fourth for Spurs, straight from a corner, with two minutes to go.
That was ultimately the goal that settled the tie, although Jonny Evans’s stoppage-time header ensured we pushed the home side right until the end.