Dyke told Sky Sports News that Hodgson has the full support of the governing body and will remain in charge of the national team, despite defeats in the opening two matches at the World Cup.
Losses at the hands of Italy and Uruguay have seen England depart the tournament at the group stage but the FA's full backing means Hodgson will remain in charge through to the 2016 European Championship finals in France.
Dyke said: “Everyone is disappointed but we are supportive of Roy. He came to do a four-year cycle and we want him to stay and continue.
“We think there are positives, despite losing narrowly to sides in the top eight (of the FIFA ranking). We don’t see any value in changing Roy. We think Roy has done a good job.
“We have young players who we’ve blooded by experiencing tournament football and it’s my view, as well as others in the FA, that he remains manager.”
And he refused to blame the multi-national nature of the Premier League as being a contributory factor into England’s poor results at the World Cup.
Dyke said: “There are not as many English players playing in the Premier League – or even the Championship – as there used to be.
“The choice is less but that’s what we’ve got to go from and I don’t think that’s the cause for what happened here.
“We’ve had narrow defeats, we haven’t been humiliated, but it’s for the football people and not me to identify why we didn’t win.”
Hodgson stressed after Thursday's 2-1 loss to Uruguay that he had no intention of stepping down as manager.
He said: "I'm bitterly disappointed, of course, but I don't feel I need to resign, no."