England were the second highest scoring side during qualification, but have struggled to sparkle at the tournament so far, finding the net just three times in their Group B fixtures.
Of the teams left standing, only four scored fewer than England in the group stage and Rooney has demanded an improvement before their knockout clash with Iceland.
"We need to play at a high tempo," Rooney told journalists. "It's a great story for them, but we have to be ruthless and try to punish them."
England enjoyed the lion's share of possession in their three group games, but were frustrated by opponents who defended deeply and with discipline, and are expecting a similar scenario against an organised Iceland side.
"I've seen them a while ago when they played Holland and they were organised and difficult to break down," Rooney added.
"Whether they sit back or have a go, it's the business end and we have to be more ruthless."
Iceland marked their maiden European Championship appearance by finishing second in Group F with four points, and coach Hodgson believes he knows exactly what to expect from the tiny island nation in the last 16.
"Since they've been here they've had some remarkable successes so we know it won't be an easy game," Hodgson told the Football Association website.
"I fear it won't differ enormously from our group games because Iceland aren't an expansive team.
"I know the coach Lars Lagerback very well and he always prepares his teams extremely well in that way, so I don't think we're going to benefit from lots and lots of space in the game.
"You take what you get in the last 16 and we're just pleased to be there, so we need to make sure we do enough preparation for the game."
(Reporting by Ed Dove; editing by Clare Lovell)
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