Draxler had come on only two minutes earlier, just days after he became the youngest Schalke player to compete in the Bundesliga at the age of 17.
Schalke are desperate to progress in the competition to secure a European spot after dropping down the league table following a bad start to the season
"I cannot describe in words what I am feeling," a beaming Draxler told ZDF Television.
"To come on five minutes before the end in a stadium where you have always dreamt of playing and to score such a goal with such a shot in the last minute is just amazing."
Holders Bayern Munich travel to Alemannia Aachen, second division Duisburg host Kaiserslautern and Energie Cottbus entertain Hoffenheim in the other quarter-finals on Wednesday.
Nuremberg shocked the hosts with a fourth minute goal when Julian Schieber slotted home from inside the box.
Schalke reacted immediately, taking control of the game and levelling 10 minutes later when Christian Pander cut a pass back into the box and little-used striker Mario Gavranovic fired home with a well-timed shot.
But Schieber, on loan from VfB Stuttgart, struck again against the run of play in almost identical fashion in the 31st minute.
The hosts stormed forward after the break, grabbed the equaliser through Ivan Rakitic and launched waves of attacks but could not deliver the knock-out punch until Draxler shook off his marker, turned and fired in an unstoppable left-foot shot.
"This was a very important step for us," Schalke coach Felix Magath said. "As for Julian, he is pure class. A great boy who has a brilliant future."
