The Turkish side went down 5-3 on aggregate to Jose Mourinho's team in their quarter-final tie, but gave them concerns when they scored three second-half goals in their game this evening.
"After the result of the first match, we didn't have much hope," Elmander told reporters.
"It's almost impossible to eliminate a team like Real Madrid coming from 3-0 behind. We showed that we are a good team with our fighting spirit tonight.
"We're proud of ourselves following our performances in this Champions League campaign."
Galatasaray lead the Turkish Super Lig by four points ahead of Fenerbahce.