This despite Hodgson's side crashing out from the World Cup after losing their first two matches, and with Italy losing to Costa Rica on Friday night. That result made it impossible for England to qualify and either Italy or Ukraine now face the same fate.
Italy's defeat is one of the shock's of the tournament and the 20/1 outsiders look like winning the group
Dyke said: "We are supportive of Roy - he came to do a four-year cycle. I already see people asking the question, 'Will he stay?' The answer is yes.
"He came to do a four-year cycle leading up to Euro 2016."
Hodgson was appointed shortly before Euro 2012 following Fabio Capello's shock decision to quit.
Dyke acknowledged it was "disappointing" England had lost 2-1 to both Italy and Uruguay, but said there were "some positives".
He added: "There are a lot of young players who got blooded here and got into tournament football."
Former England striker Alan Shearer had earlier told BBC Sport the FA should stand by 66-year-old Hodgson.
He said: "I would give Roy the chance to take England to the 2016 European Championship. He should get the opportunity to finish what he has started."
It is the first time England have failed to get out of their group in Brazil since the World Cup finals since 1958.
But Shearer, 43, believes Hodgson has done what was asked of him and does not think there is a better candidate for the job.
"If we do sack him, who would we replace him with? Roy is a good manager and an experienced manager and I don't think anybody else would have done anything differently at this World Cup.
"He has picked young players, which is what everyone wanted him to do, and the fact of the matter is that we have not been good enough.
"That is the issue and a new manager would have the same problems, regardless of who it is."