The Turkish club were 3-2 winners in Tuesday night's return game in Istanbul, but they ultimately crashed out of Europe after a 5-3 defeat on aggregate.
Eboue netted the equaliser in the second leg, but he remains disappointed by their Santiago Bernabeu defeat.
"We let ourselves down in the first game, we conceded goals cheaply there and that has shown why we are out," he told Sky Sports.
"We played very well; we showed we are not far from them. Last week we showed too much respect, we had the right spirit .
"Now we will need to finish the season strongly and be in the Champions League next season. We will improve, I am sure of that."