The Bundesliga club trailed 3-0 from the quarter-final first leg at the Bernabeu last week, but stunned the Spanish giants by racing into a 2-0 half-time lead in Tuesday's second leg at Signal Iduna Park after Real's Angel Di Maria had missed an early penalty.
Dortmund midfielder Henrikh Mkhitaryan struck a post as Jurgen Klopp's men attempted to take the tie into extra-time, but Real held on to progress to the semi-finals.
Reus, who scored both goals, was left to rue a disappointing performance in Madrid but felt Dortmund had more than enough chances to see off Carlo Ancelotti's side.
He said: "We didn't play well in the first leg and that's why we were chasing the tie. We did everything we could right from the start and played with real courage going forward.
"Unfortunately we just lacked that little bit of luck, but we came mighty close to surprising everyone.
"We could easily have scored the goals to take us through, both tonight (Tuesday) and last week. That makes it all the more difficult to swallow that we're out. After reaching the final last season we definitely wanted to get further than the quarters."
The result saw Real gain a measure of revenge for last year's semi-final loss to the same opponents.
