Walcott came off the bench to inspire England to victory.
"Theo's contribution was enormous," said Hodgson.
"To come on as a sub, score and set up the winner means he'll be very happy.
"Of course I believe in him.
"He's a good player with good qualities. That's why I selected him in my 23.
"I have only worked with these guys for a month and am still learning what they're really good at, and what areas we can find improvement.
"But I was delighted with his performance tonight."