Daniel Sturridge scored the only goal of the game with a header eight minutes from time after England had looked set to fail to win at Wembley for a third consecutive game - following defeats to Germany and Chile.
The outstanding Kasper Schmeichel produced several fine saves to thwart the likes of Sturridge and Danny Welbeck before the Liverpool striker nodded in Adam Lallana's cross.
Despite having several chances and looking the more dangerous side, there is still plenty of room for improvement for England ahead of this year's showpiece in Brazil.
And after seeing Roy Hodgson's side at close quarters, Olsen is not expecting them to be among the contenders to lift the trophy in South America.
"I wish you (England) good luck in the World Cup. You need it," he said.
The friendly was Denmark's first international since Olsen signed a new two-year contract and the long-serving coach admitted it was not a pretty spectacle.
"It was not such a good football game, I have to be critical," he added.
