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Christian Gentner and Martin Harnik had put struggling Stuttgart 2-0 ahead inside 20 minutes at the Mercedes-Benz Arena on Saturday, but a Marco Reus hat-trick inspired the visitors to a 3-2 win.
Dortmund head coach Klopp will now prepare his team for the UEFA Champions League quarter-final first-leg against Real at the Bernabeu in midweek.
"We got into this round without winning in the Bundesliga beforehand," he said.
"Either way, we will face huge problems in Madrid like any other football team that has been there so far this season.
"Nevertheless we were very happy when we qualified for the next round. That the opponent was going to be a strong one was to be expected. We won't make it easy for them.
"We are definitely not massive favourites but we want to get a result that will give us the opportunity to do something in the second leg.
"We just have to recuperate now, the boys were really tired at the end of the game (against Stuttgart), it was hot and a real intense game for everyone."
