The FA Chairman Greg Dyke confirmed on Friday that Hodgson will stay on in the role to lead the Three Lions into Euro 2016.
And speaking in Rio this afternoon, Rooney says he agrees with the decision and believes the national team is making progress.
Rooney celebrating his first World Cup goal
"He's put enthusiasm in the team. There's no doubt Roy is the man to take us forward," said Rooney.
“He's changed the way we play over the last couple of years. You can see the progression. I feel we're getting better and this will help the younger players and help us become a better team."
Having lost their opening two group matches against Italy and Uruguay, England’s fate was confirmed on Friday when Costa Rica defeated the Italians 1-0.
They will complete their campaign on Tuesday when they face the Costa Ricans on Tuesday at 5pm (BST).
Despite being knocked out, Manchester United striker Rooney believes that the talent being produced in the country will make the national side a competitive force on the international scene in the future.
The 28-year-old, who scored his first World Cup goal in the Uruguay defeat, said: “If you look at the Premier League over the last couple of years, the number of English players coming though is incredible.
"Ross Barkley, Luke Shaw and Raheem Sterling have had outstanding seasons and they've progressed so quickly, playing week in week out for their clubs and stepped up at international level.
"So we have to be excited about that and hopefully in the future there will be more like that in the national team.”
