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 left the Irish without a point to their name from their three matches in a tournament, in which their defending let them down.
Spain's 1-0 win over Croatia in Gdansk meant the reigning champions topped Group C on seven points, while Italy went through in second place with five points.
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 said. "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 last night 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."
On a night when only victory would do for Cesare Prandelli's men, they came up with the goods against an Ireland side whose exit from the competition was confirmed last week.
"It was a very difficult match, we came up against a team that made us suffer," Prandelli said. "We knew that heart mattered more than quality. Last night we created a lot, I hope it will be like this also in the next match."
Having conceded early goals against Croatia and Spain, Ireland were determined not to get caught out again and went straight on to the attack.
Backed by a vociferous and large contingent in the crowd, they looked briefly threatening against Italy.
But it was Cassano, who missed half of the season with AC Milan after undergoing a minor heart operation, who headed home 10 minutes before the break from Andrea Pirlo's inswinging corner.
Italy kept attacking after the break, Cassano wasting a chance with a side- footed shot before Daniele De Rossi curled an effort over.
A neat combination between Thiago Motto and Cassano then set up Antonio Di Natale, whose shot from a difficult angle was blocked by keeper Shay Given.
Keith Andrews reminded Italy that the game was not over as his awkward, long-range shot gave Gianluigi Buffon his first real test of the game on the hour.
Italy began looking nervous and there was a rare moment when Pirlo lost possession to Andrews in midfield, but De Rossi came to the rescue with a timely tackle to block the midfielder's shot.
Another thumping Andrews shot was saved by Buffon as Italy clung on and the Ireland midfielder was sent off for his second book-able offence moments before Balotelli settled matters with a superb piece of skill, the maverick forward living up to his reputation when he failed to celebrate a stunning goal.