The Turkish outfit, trailing 3-0 after the first leg of the UEFA Champions League quarter-final, fell further behind when Cristiano Ronaldo scored after eight minutes before three second-half goals raised hopes of an unlikely comeback.
Ronaldo struck again near the end to end Galatasaray's hopes, but their battling performance in the 5-3 aggregate defeat pleased the Sweden international.
"After the result of the first match, we didn't have much hope," said the former Bolton striker.
"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.
"We've had a good season in this competition after reaching the last eight. Now we want to concentrate on the league and finish by winning it."