With the scores level at 2-2 and 3-3 on aggregate after the first leg in Amsterdam, Babangida grabbed a dramatic winner for the Dutch side, who now meet Juventus in the semi-finals next month.
Edwin van der Sar's penalty save from Juan Esnaider 15 minutes from time earned his side extra time, with the score level at 1-1 thanks to goals from Kiko and Ronald de Boer. Dani, the former West Ham forward, put Ajax ahead, but Milinko Pantic levelled from the spot.
Juventus, who beat Ajax on penalties in last season's final in Rome, reached the last four by beating Norway's Rosenborg Trondheim 2-0 to advance 3-1 on aggregate. Zinedine Zidane and Nicola Amoruso, from the penalty spot, were on target.