The Arsenal winger has not started any of the four games since Hodgson replaced
He came good when it mattered though, coming off the bench to inspire a famous England comeback as the Three Lions beat
Walcott scored a brilliant equaliser within three minutes of his arrival, then set up
"Of course I believe in him," said Hodgson.
"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."
England now requires a point from its final Group D outing against co-host
And, even though Welbeck was joined on the scoresheet by fellow striker
"I welcome those kind of selection problems," said Hodgson.
"I was really pleased with the performance of the front players.
"I'm afraid you expect him to get back into the team when the suspension is served."
The statement merely underlined what a massive role Rooney is being afforded in the Hodgson era.
He must have gone through every emotion in the stands as England established a winning position through Carroll, tossed away the initiative with some chronic defending to gift
"It was a good move," said Hodgson of the winner.
"Theo was able to skip away from the full-back and showed a lot of composure to find Danny at the near post.
"He timed his run very well and it was a very clever finish with the pirouette past the goalkeeper who thought he was going to toe it towards the near post.
"It was a great goal to win the game."
Arsenal midfielder Jack Wilshere hopes the latest operation on his troublesome ankle will prove the last as he looks to recover full fitness for both club and country next season.