England's opening game of the tournament ended in a 2-1 defeat in Manaus, with former Manchester City striker Mario Balotelli scoring a 50th-minute winner.
Roy Hodgson's side now have it all to do in their remaining Group D games against Uruguay on Thursday and Costa Rica next Tuesday if they are to progress.
Manchester United forward Rooney said: "We always believed we could win the game and getting the equaliser put us on the front foot going into half-time.
"But, going 2-1 behind, we knew it was going to be tough then. Italy are a good team and they make it difficult for you.
“I was disappointed that I didn't score but we will move on. We are looking forward to the game on Thursday and hopefully we can get three points."
And Chelsea defender Cahill echoed captain Steven Gerrard’s sentiments that the quality of England’s performance was beyond doubt, despite the result.
“We started the game really brightly, did well and there wasn't too much we did wrong,” he said.
“Unfortunately we were on the losing team, but two good teams went at it. They have some very good players.
“Sometimes, it's cruel.
"But we can take a lot of positives from that, I thought we played well.
"We had more joy in the second half. We had more possession, we were well in the game, and I thought they found it difficult in the second half.
"We had a feeling out there that, if we'd scored, we could have gone on."
Right-back Glen Johnson said Italy are strong opponents, who are always difficult to overcome.
"Italy are set up to be hard to beat," he said. "But we didn't deserve to lose."
England will now face Uruguay in a massively important match in Sao Paolo on Thursday.