The result means that England face an uphill battle to reach the knockout stages, but Dyke has confirmed that the manager will receive the FA's full backing.
"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 told BBC Sport.
"He came to do a four-year cycle leading up to Euro 2016."
Hodgson has confirmed that he will not resign from his position.