English FA chairman Greg Dyke had said before the tournament that making the semi-finals or being eliminated at the quarter-final stage by "a good team" would be considered a success, but Rooney has set his sights considerably higher.
"We are here and we want to win it," England's all-time leading goalscorer told reporters.
"Whether it happens remains to be seen, but we're not going to say that getting to the quarter finals will be a sign of progress. I believe we are better than that."
England have not got past the quarter-finals in their last three European Championships and failed to make it out of the group stage at the World Cup in 2014, but Rooney sees cause for optimism this time round.
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"If this was four years ago and you were saying you have to play France, Spain and Germany you would have been worried," he added.
"I think the gap has changed, and not just with ourselves, but with the likes of Wales. The gap to get to those teams is not as big."
Manager Roy Hodgson's men are aiming for their first major international title since the 1966 World Cup, but will have to find a way past an inspired Iceland in the round of 16 game on Monday. They could face France for a spot in the last four.
Hodgson took six strikers to France, but his team have struggled to turn dominance into goals, scoring three times in three group games and winning only one.
Despite the team's struggles in the final third, Rooney, who has been pulling the strings from a deeper midfield position, said the talent in the side meant he felt less pressure to deliver.
"I have always gone into a tournament thinking if I don't play at my best I cannot see us winning it," said the player who burst on to the international stage as an 18-year old with four goals at Euro 2004.
Rooney has since failed to make a telling impact at the continental championship and though he is yet to open his account in France, he is pleased that the England squad have the talent to ease the burden of expectation on his shoulders.
"I have come into this tournament and we have several players capable of doing magical things. We have five or six match winners in our team and I cannot say we have always had that," he said.
(Reporting by Nivedita Shankar in Bengaluru, editing by Ed Osmond)
