Hodgson failed to mastermind a first semi-final appearance since 1996 as his troops bowed out at the last-eight stage yet again as Italy prevailed 4-2 on penalties.
The former Fulham, Liverpool and West Brom boss has only been in charge for just over a month and Redknapp feels that people need to give him time to show what he can do at this level.
Redknapp told Absolute Radio: "No-one's blaming Roy or anyone else, that's for sure. I mean he's just took over, he's going to have to get his own style of play, but we've got to start playing some football haven't we?"
The former Spurs boss refuses to believe that England do not possess enough quality players to win international tournaments, and urged them to step up.
He said: "It is unbelievable. We keep underestimating our players. I don't care what anybody says, we have got good players.
"Our back four and the goalkeeper will be as good as any back four and goalkeeper in the tournament, man for man. You know, Glen Johnson, Ashley Cole, Joe Hart in goal, John Terry. Joleon Lescott had an excellent tournament.
"You've got Steven Gerrard, you had Scott Parker, you've got Wayne Rooney coming back."
Redknapp, though, is under no illusions of the fact that England must improve the technical aspects of their game, as was demonstrated on Sunday night as Italy out-passed Hodgson's men in a one-sided affair.
"We can't keep giving the ball away in international football and expect to win games," Redknapp added.
"I mean our possession was less than any team in the tournament I think, apart from the Irish. That can't be right. It's not as if we can't pass the ball."