The 30-year old Australian goalkeeper has impressed this season when deputising for Anfield number one Pepe Reina and has been rewarded with a long-term deal.
"I said when I signed here it was a dream come true," he told the club's official website.
"It was something I'd wanted as a young boy - and that hasn't changed. I want to stay here as long as I can.
"At the minute, the club is on a progression and we're improving a lot. It's a good time to be here and I'm sure in the next couple of years, I'll be part of a successful squad."
He added regarding Rodgers: "He's a big influence. When he came in, I don't think he was too sure about whether I was going to stay or not and I had to prove to him I was worthy of being here."