The Russians took a one-goal deficit into their home leg but soon fell further behind when Wesley Sneijder scored for the Italians.
Then to make matters worse, CSKA had winger Chidi Odiah sent from the pitch just after half-time which ruled out their chances of winning but Slutsky said his players could hold their heads high after being one of the top eight teams in Europe this season.
"We came into the match looking for a win. But the early goal, along with Vasili Berezutsky's injury made our task a great deal tougher," said Slutsky.
"The dismissal of Chidi Odiah dashed our hopes almost completely. But I can't hold anything against my men today. They did all they could."
He added: "I consider nevertheless that we looked good in both of our matches with Inter and deservedly had a place in the last eight."