Mario Balotelli put Clarence Seedorf's side in front in their quarter-final against Udinese at San Siro, but the visitors progressed courtesy of a Luis Muriel penalty and Nicolas Lopez's winner 12 minutes from time.
It was a first defeat for Milan under Seedorf as their miserable season took another turn for the worse.
The Dutchman's team are 11th in Serie A, and although they are still in the UEFA Champions League, it would be a major surprise if they were competing in Europe next season.
However, Brazil international Kaka is determined to use Milan's struggles as motivation to turn their fortunes around.
He told Milan's official website: “We’re really disappointed. The cup was the best way to get into Europe next season and now we have a tough road ahead of us in the league.
"All credit to Udinese but once we took the league we should have got the result. The formation didn't work but against Verona (in a 1-0 victory on Sunday) it did.
"We'll start off again from that and keep working. We have little to talk about and lots to work on.
"It's a tough moment and it's bad to get knocked out of the cup at home. There's a chance now that we won't be playing in Europe next season but we should take this as a motivation.
"It's hard to imagine a Milan side that doesn't play in Europe. It's tough for all of us at the moment. We have a lot of improving to do."
