Brazilian Lucas Moura scored a second-half hat-trick to secure a 3-2 win in Amsterdam and send Tottenham through on away goals to the final in Madrid on June 1.
Trailing 1-0 from the first leg, Spurs looked out of it at half-time after a towering header from Matthijs de Ligt and a curling finish from Hakim Ziyech.
But Lucas stuck twice inside four minutes to level the scores early in the second half, first slotting home a brisk counter-attack before wriggling past two defenders in the box to smash in the equaliser.
He completed his remarkable hat-trick when he scored the dramatic winner deep into stoppage time to seal another historic night in this competition, and leave Spurs boss Mauricio Pochettino in tears.
Spurs will now face Liverpool in the final in Madrid on June 1.