Dyke, who famously made a throat-cutting gesture after the final draw in December, said the opponents and the venue were just the ones England had wanted to avoid.
Speaking in Sao Paulo, he said: "The gesture was a joke. It was a joke, and you sometimes have to accept there's a sense of humour in the world.
"We sat there and said there were two things we didn't want. We didn't want Italy and we didn't want Manaus, and we got both!"
Dyke would not be drawn on a minimum requirement for Roy Hodgson's side or on the manager's future - Hodgson is contracted until the end of Euro 2016.
He said: "I've got a lot of time for Roy. I'm a big fan.
"We wouldn't want to discuss what happens to him afterwards because this is the moment when things should be going well. So it's not appropriate to discuss that now. But I'm a big fan of Roy's."
Asked for England's minimum requirement, Dyke replied: "I would say... we would obviously hope to get out the group and then after that...
"It's a tough group. I think one of the reasons we are coming here with less expectation is because it's such a tough group.
"You compare with the others. I was looking at the French group the other day. If the French don't get out of their group I think they will be committing suicide. We are in a tough group and a lot of it is going to be about that first game."
Hodgson, meanwhile, thinks Dyke was wrong to make a cut-throat gesture when England were drawn in their so-called 'Group of Death'.
Hodgson, who was sitting next to Dyke when he made the gesture, does not share such a pessimistic outlook on England's chances.
And although he tried to play down the incident at a press conference in Rio on Tuesday, there was no hiding the fact that Hodgson thought the gesture was ill-advised.
"I didn't see it (at the time), I was sitting next to him staring straight ahead, and I've never spoken to him about it," the England manager said.
"Do I think it's a difficult group? Of course. Should one make a gesture like that? I presume not.
"It seems such a minor thing."
Hodgson himself caused controversy prior to the draw in Salvador when he said Manaus was "ideally the place to avoid" in the World Cup due to the local humidity levels, which can hit 90 per cent.
As luck would have it, England were then drawn to play there against Italy in their opening match of Group D.
Hodgson's remarks did not go down well with Manaus' mayor, who said England were "not welcome" in the jungle city.
The England boss, who has since visited Manaus to smooth things over with the mayor, denied he was dreading the trip to the Amazon, where they fly on Thursday 48 hours before taking on the Italians.
"It is not true that we complained about Manaus and Brazil," Hodgson said.
"Quite the opposite. I am more than happy to be here now.
"We are very much looking forward to playing in a World Cup and we will be perfectly happy to play in any city we are given."
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- Roy Hodgson