The Poland international struck four times during the game to give his side a strong lead going into the semi-final second leg at the Santiago Bernabeu next week
The 24-year-old netted the opening goal of the semi-final first leg in the eighth minute, before adding three further strikes after the break.
"I’m happy to have scored four goals, but obviously the big picture is reaching the final and that’s the only thing that counts," he told Uefa after the final whistle.
"Real played well in the first half but we had things under control in the second half.
"We are very satisfied."
Midfielder Ilkay Gundogan spoke of his pleasure at being part of the victory, particularly as it was against such talented opposition.
"It was a royal evening for us, it would have been perfect if we had kept a clean sheet," he added.
"However, it’s quite hard to get a better result against a team like Real.
"We played a great game from the start until the final whistle and deserved to win by this margin."
Dortmund's sporting director Michael Zorc was also happy with the German club's performance, and he reserved special praise for Mario Gotze, who recently agreed a move to Bayern Munich which will be completed in the summer.
"It was an extraordinary performance, for almost the whole 90 minutes," he said to Sky Sports.
"The goal we conceded was our mistake, but the team corrected itself in the second half.
On Gotze, he said: "Gotze was very good and very concentrated, just like every other player."
The second leg will be played at the Santiago Bernabeu in Madrid on April 30.