Rooney's 48th minute header against Ukraine sealed England's qualification into the quarters for a clash against European giants Italy at Kiev's Olympic Stadium.
The Manchester United striker, who returned after serving a two-match suspension, was relieved to have scored the winner and his first goal in a major tournament for eight years.
Having lost in three quarter-finals and a departure in the last 16 at the 2012 World Cup, Rooney feels England have the ability to reach the semi-finals.
"I hope there is the belief we can go past the quarter-finals this time," Rooney said.
"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."
"I haven't scored at a major tournament for a while so it was nice to get a goal which won us the game. We know our qualities and what we're capable of. We've worked hard, done a lot of tactical play in training. We're difficult to beat and I don't think any team will fancy playing us," Rooney said.
England captain Steven Gerrard, who has been the most influential player with three assists in three games, said teams need a bit of luck in major tournaments when pointing out Marko Devic's shot which had crossed the line but denied by the referee.
"To be successful in these tournaments, because of the standard of teams involved, you need that bit of luck going with you," he said.
"Two years ago we didn't get that luck with Frank Lampard's goal, a big turning point in that World Cup game against Germany, and we ended up packing our bags and going home.
"Against Ukraine, the luck turned. But it just shows that, if you keep working hard and keep fighting, you earn that luck. We deserved it," Gerrard said.