Antonio Cassano headed in from a corner to give Italy the lead in the first half before Mario Balotelli sealed Italy's place in the quarter-finals with a late strike.
It leaves the Irish without a point to their name from their three matches in a tournament where their defending has let them down.
Keane paid tribute to the travelling fans, but admitted they had been one of the few bright spots of the competition.
"The last three games, they've been fantastic," he told BBC3.
"It's a shame we didn't perform as well as we can and perform for them because if anyone deserves it, it's them.
"I think we played well without really creating too many chances. It's one of those things. It's certainly disappointing we didn't get something out of the group.
"We're all disappointed, but we've been beaten by better teams."
Keane pointed the finger at his side's defending as they conceded too many soft goals.
"It's set-pieces again and it's been the same in the last few games. It's one of those things because in previous games we've been quite solid in that department."
Asked if this was likely to have been the final appearance for several of Ireland's veteran players - including Keane himself - the 31-year-old Los Angeles Galaxy forward said: "We'll have to see."