Poland striker Robert Lewandowski netted all four of Dortmund's goals to strengthen the possibility of an all-German final at Wembley on May 25, but Cristiano Ronaldo's away goal gives Real a slim second-leg life line.
After Bayern Munich's 4-0 hammering of Barcelona on Tuesday in the other semifinal, Borussia followed the German champions' example and floored their Spanish opponents at Dortmund's Signal Iduna Park.
Mourinho's team now face an uphill battle at the Estadio Santiago Bernabeu in Tuesday's return leg, but with Real chasing a tenth European title, the 50-year-old self-styled "Special One" has promised they will put up a fight.
"I have learnt that anything can happen in football, it will be tough next week but nothing is impossible and my footballers will try to show that next Tuesday," said Mourinho.
"The best team by far won. Individually and collectively Dortmund were better and deserved their victory.
"They were first to every ball and were far more intense than us. We found it hard to hold onto possession when they pressed, but we can come back."
Real only narrowly avoided suffering their worst defeat in international competition – AC Milan beating them 5-0 in what was then the European Cup in the 1988/89 campaign – but Dortmund coach Jürgen Klopp advocated caution.
'No walk in the park'
"We have to force ourselves to stay focused, we still need to go to Madrid, which will be no walk in the park," said Klopp.
"We scored some great goals and played really well, but we have to go to Fortuna Düsseldorfer, in the Bundesliga, then Madrid for the second game and there is still a lot to do."
Klopp quickly poured water on any notion his side are now favourites to win the Champions League for the second time in their history after their 1997 triumph.
"I don't have the feeling we're favourites, we produced an exceptional game against a fantastic team, there could have been worse results and we'd still have to go there, we just want to reach the final," said Klopp.
"Nothing is decided, it's just half-time, we don't have much experience this far in the tournament and we will have to work unbelievably hard in Madrid.
"The only thing I know, which is great, is that my team will not go there and want to defend."
Götze to join Bayern
Klopp had promised his players would not be destabilised by the news on Tuesday that 20-year-old midfield star Mario Götze was to join Bayern Munich next season and his prophecy proved correct.
The night belonged to Lewandowski who stole the limelight with the opening goal, then added a second-half hat-trick to became the first player to score four goals in a Champions League semifinal.
"We have taken only the first step, but of course we are happy," said Lewandowski.
"I am pleased with the four goals, but now we want to reach the final."
With Lewandowski having now scored 34 goals in all competitions this season, but threatening to leave when his contract expires in June 2014, Klopp uttered the words all Dortmund fans want to hear.
"He's a quality player and I am sure he will stay." – AFP