They will play hosts France in the quarter-final on Sunday after goals from Ragnar Sigurdsson and Kolbeinn Sigthorsson knocked out England and forced manager Roy Hodgson to resign.
Iceland, ranked 34th in the world, were surprise qualifiers for what is their first international tournament.
"We all believed. The rest of the world didn't but we did," said Arnason.
Iceland's population is just 329,000 - not many more than live in Coventry and almost 10 times fewer than Wales - and it is believed 8% of those are in France following their side.
The island nation, which has no professional clubs, ranked 131st in the world during the last European Championships four years ago.
"This is without a doubt the biggest result in Icelandic football history. We've shocked the world," added Arnason.
Wayne Rooney put England ahead with a fourth-minute penalty but Iceland were ahead by the 18th minute and held on comfortably despite Hodgson's side having 18 attempts on goal.
Iceland defender Sigurdsson, 29, accused England of complacency against a side who had already held Portugal and Hungary and beaten Austria during the group stage.
"They thought this would be a walk in the park but we had faith in our ability," Sigurdsson said.
"It went well. We didn't feel that England created any chances. We were just heading away long balls. I wasn't stressed in the second half."
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Iceland coach Heimir Hallgrimsson - joint manager with former Sweden boss Lars Lagerback - praised his side for taking their chances and is in confident mood for Sunday's quarter-final in Paris.
"I was more relaxed than during the game against Austria," he said.
"If someone had told me a few years ago that we would reach the last eight, I have to say I would not believe it.
"But no obstacle is too big for these guys now. If you want the best out of life you have to be ready when the opportunity comes. That is a fact and these boys were ready and this opportunity that was put in our hand was huge - it can change their lives.
"We are optimistic. Some Icelanders maybe think we are too optimistic, that we don't think we can fail, but we have a gameplan."
Icelandic TV commentator Gudmundur Benediktsson went viral last week with his excited reaction to the win over Austria and he was at it again during the win in Nice - even seeming to reference the result of Britain's EU referendum.
"This is done! This is done! We are never going home! Did you see that! Did you see that!" he said.
"Never wake me from this amazing dream!
"Live the way you want England! Iceland is going to play France on Sunday. France-Iceland! You can go home. You can go out of Europe.
"You can go wherever the hell you want. England 1 Iceland 2 is the closing score here in Nice. And the fairytale continues."
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Police said a number of charities and businesses in counties Down and Armagh were targeted by men passing counterfeit Â£20 notes on Thursday.
It involved premises in Banbridge, Dromore, Ballynahinch, Moira and Newtownhamilton.
Four men, aged 20, 22, 31 and 45, were arrested on suspicion of fraud in Newtownards town centre.
They were detained when officers on patrol in High Street stopped a black vehicle after they noticed "suspicious" activity.
The men are currently assisting police with their inquiries.
Iceland's European Championship last-16 win over England "shocked the world", according to defender Kari Arnason.
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Four men have been arrested in Newtownards, County Down, in connection with a fake banknotes investigation.