Jurgen Klopp’s side were beaten 2-0 on the night at the Santiago Bernabeu but will contest the final at Wembley on May 25 after prevailing 4-3 on aggregate.
Late goals from Cristiano Ronaldo and Sergio Ramos were not enough after BVB ran out 4-1 winners in Dortmund last week and the Poland international has said that the team should celebrate in the immediate aftermath of the result.
"We can rejoice that we have made it to the finals," Piszczek told reporters after the game.
"It was not an easy game, as we had predicted it wouldn’t be directly after the first leg. Real can always score three goals at home but we’re through and we can be happy.
"We had the idea to go to Madrid and score another goal, which would have made things easier and it was not easy after [Ronaldo’s] opening goal. Fortunately, they did not score again [after Ramos’ strike]."
The 27-year-old also played down suggestions that he had the measure of Ronaldo after playing against him three times this season and pointed to the Portuguese forward’s two goals over both legs as evidence.
"I must also say that my whole team has helped to get him under control. One player cannot completely keep him quiet. He has after all, two goals over the three games, so I would not say that I have always kept him out of the game."
Dortmund will play the winners of Wednesday’s tie, which pits Bayern Munich - who have a 4-0 advantage - against Barcelona.