Jordan Jewson looked to have secured progression for the hosts with a goal just after half-time in extra-time - but Ross Wilkinson's strike means the two sides will meet again tomorrow night.
A gripping round 2 qualifier required extra-time before the sides were separated - and the Deise could avenge last year's heartbreaking All-Ireland semi-final defeat.
Former Manchester United manager Sir Alex Ferguson, UEFA's coaching ambassador, said he did not like to see players "walking" around the pitch exhausted and that extra-time rarely succeeded in producing a winner.
Jordao equalised for Portugal to take the game into extra-time and the Sporting forward would give the outsiders the edge by adding his second after 98 minutes.
After the match ended goalless in 90 minutes, midway through the second period of extra-time Gotze controlled Andrea Schurrle's cross on his chest and superbly volleyed past Argentina keeper Sergio Romero to give Joechim's Low's side a 1-0 victory.