The Italians won 4-2 on penalties aftr the match, dominated by Italy, finished goalless at the end of extra time.
"I don't think I could ask any more than the players gave tonight," Hodgson told a news conference.
"They really gave their all. We weren't quite good enough to win it over the 90 minutes and when it came to penalties we went down the same road we've gone down so many times before," Hodgson added.
England went into penalties hoping to improve on a record that has seen them only win one shootout in a major tournament, against Spain at Euro 96.
But eight years to the day after Portugal knocked them out of Euro 2004 in the same manner, Ashley Young and Ashley Cole missed from the spot and England exited another tournament.
"Once again we go home because we can't win a penalty shootout, which quite frankly isn't the same as a game of football," Hodgson said. "It's an entirely different competition."
The England coach sympathised with Cole and Young, but praised the nerve of Italy's Andrea Pirlo, whose chipped effort fooled England goalkeeper Joe Hart.
"The cool, calculated way that Pirlo had to chip the goalkeeper, that is either something you have or you don't as a player," he said. "You can't coach that."
With Pirlo majestic in the centre for Italy, Hodgson told reporters that his central midfielders, Scott Parker and Steven Gerrard, were suffering from injuries.
"Scott was feeling his Achilles [tendon] and Steven was struggling with cramp," the manager said.
Hodgson said both sides had had chances to win the game.
"I think there was chances missed at both ends. Both teams created chances, but I think Italy played well. I wouldn't begrudge them their victory," Hodgson said, adding that it was an "interesting tactical battle" between him and Italy coach Cesare Prandelli.
"Talking for my own team, I though we put up a very, very good effort and had we won it we wouldn't have been apologising for the victory," he said.
"I've been very happy with the way they've responded to the demands of an England shirt. They've shown they're well capable of stepping up to the plate."
