Tottenham looked to have put themselves in the box seat when Christian Eriksen's deflected effort opened the scoring, only for a fine Mauro Icardi volley to bring them back into the game late on.
The hosts threw everything forward in search of the win as the game moved into stoppage time and their bravery was rewarded in the second of four added minutes.
A corner from the right was headed back across goal by Stefan de Vrij, with Vecino reacting quickest to nip in ahead of Harry Winks and plant a header into the back of the net to earn his side all three points.
The goal sparked scenes of jubilation at the San Siro as the 'Nerazzurri' made a winning return to Europe's premier club competition.