Galatasaray's 3-2 victory at the Turk Telekom Arena proved to be insufficient after the 3-0 loss at the Santiago Bernabeu last week and Drogba once more rued Cimbom's first-leg performance.
"It was a really good game but it would have been even better if we'd managed to qualify. If we'd played like that in the first leg the score might have been different," Drogba was quoted as saying on the official UEFA website.
"We saw again tonight that the fans here are extraordinary. It's a shame but I think Madrid deserved it. Them scoring so early set us back but in the second half we showed after scoring that we believed in ourselves."
The Cote d'Ivoire attacker netted once on Tuesday but he stressed that he would have preferred to progress to the next round rather than score.
"[My goal] is not so important because we are out. I don't like to score and be out - for me it doesn't mean anything," Drogba explained.
Galatasaray resumes domestic action against Kardemir Karabuk at the weekend.