England’s victory over the co-hosts, coupled with Sweden’s 2-0 triumph over Laurent Blanc’s France, saw Hodgson’s men top Group D and set up a quarter-final clash with Italy at Euro 2012, avoiding world and European champions Spain.
After missing the draw with France and the 3-2 triumph over Sweden through suspension, Manchester United forward Rooney was restored to England’s starting XI where he made an immediate impact, grabbing the winner early in the second-half.
And Hodgson has praised the 26-year-old’s quality and work-rate in the victory.
"He's got character, hasn't he?" Hodgson told reporters.
"Those who have followed England and Manchester United, even Everton, you know what Rooney is, what he can do and his qualities.
"He showed them in abundance [against Ukraine]. He and [Danny] Welbeck worked extremely hard.
"It was his first game after having to sit these two others out, and it is a while since he played a competitive match, so he's got to be happy with the way things went for him.
"Every day helps him and helps us. The fact he's had 80 minutes in this game will give him confidence going into the next game."
Despite being handed the England job just over a month prior to the kick off of the European Championship, Hodgson says he is delighted with how his tenure has begun and believes the nation is beginning to believe in his side a little more.
He continued: "After Fabio Capello resigned people thought the team wouldn't have enough time with a new coach to prepare.
"Luckily we've proved that to be wrong. I'm pleased about that. As far as our stay in Ukraine and Poland, I've been delighted with that.
"Now we are getting good vibrations from those back home. It's not really a question of expectations. We want to keep going and try and enjoy this tournament for as long as we can."