The outgoing chairman of the Football Association said he felt the young England team had the potential to outplay finalists France and Portugal, despite their exit in the round-of-16 stage.
His words come a day after Sam Allardyce gave his first press conference as Roy Hodgson's successor in the England job.
New manager Allardyce suggested that the team Hodgson had going into the tournament was good enough to raise the trophy, adding that the squad's ability "seemed to dry up" as they progressed.
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Speaking outside the home of Bobby Moore, where a blue National Heritage plaque was being unveiled in memory of England's 1966 World Cup winner, Dyke told the Press Association: "We need to keep winning - and we were winning in all the group stages, we just didn't do very well in the finals."
Asked if he agreed with Allardyce's comments that Hodgson had a strong enough team to lead the country to glory, Dyke said: "Yes, I don't think Portugal were that good and I don't think France were that good - I think we could have won in 2016, but we didn't, did we?"
He added: "We've got a talented group of players at the moment and it's about making sure they perform."
Dyke, who chose not to stand for re-election after three years as one of English football's most senior figures, said the challenges facing Allardyce going forward would be to help ensure young English players break into the Premier League.
He said: "The big challenge is that fewer and fewer English players are coming forward into the Premier League and that can't go on, we've got to change that."
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