England reached the last eight in the 2002 and 2006 World Cups plus Euro 2004.
Rooney said: "I hope there is the belief we can go past the quarter-finals this time.
"It will be a difficult game against Italy and it will be quite evenly matched in terms of the way we play.
"It can go either way but we're looking forward to it. We're feeling confident and we've got players in the team who can score goals."