After consecutive 2-1 losses to Italy and Uruguay, England needed Cesare Prandelli's side to do them a favour by beating Costa Rica on Friday, but the Italians succumbed to a surprising 1-0 defeat.
Although qualification looked highly unlikely before Costa Rica's win, the result confirmed England’s elimination without even playing all three of their Group D games.
Despite the disappointing World Cup showing, Football Association chairman Greg Dyke gave Hodgson his support on Friday and the coach is confident his squad has the ability to make more of an impact in the future.
He said: "We had high hopes, we thought we could make an impact, but we haven't won the games. To make an impact you have to win games and we've lost both of ours.
"I believed the team, going forward, would be a very good team - there are a lot of good young players. Even against Uruguay we saw some good individual performances.
"But results colour everything. I don't think there is any need to have doubts that this group of players will go on to do some good things in times ahead.
"The team is evolving and I'm proud of the way they have approached this tournament, everything they have put into it.
"We've worked so hard, we've done so much preparation for this tournament, we think we came to the tournament well-prepared.
"We had really big hopes that we were going to make the country proud by going far in the tournament, but we haven't done that."
One of the players who has received the most criticism for England's disappointing performance in Brazil is Wayne Rooney and the Manchester United forward revealed his sadness at World Cup failure on Saturday.
"Absolutely devastated to be out of the World Cup," he said on his official Facebook page.
"Going into each game we had great belief in ourselves but unfortunately it hasn't worked out.
"Sorry to all the fans that travelled and at home that we haven't done better... gutted!”
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