Rodgers had initially cast doubts over his long-term future at the club.
After a string of good performances standing in for the injured Jose Reina, the Australian has been rewarded with a new deal.
But he admits when Rodgers came in during the summer the situation was much more uncertain.
"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," said Jones.
"He was very honest with me at the start and said maybe things weren't going to work out.
"But he was then just as honest to come and say to me I'd done well and proved myself to him.
"That's what you want from your manager - somebody that's honest, that you can talk to and is going to help you progress as a player.
"He's a big influence. He's a very positive guy and somebody that wants to be amongst the players, talking to them and making sure they know what he wants.
"I'm happy to stay on and happy the club have been supportive with me and I've seen enough to want to keep me here longer."
Jones has made 10 appearances so far this season, by far his best run in the side having amassed a total of just 14 since joining from Middlesbrough in 2010.
A week ago Rodgers moved to dismiss speculation he was preparing to bring in a new goalkeeper in the January transfer window having been linked with Birmingham's England goalkeeper Jack Butland and young German prospect Ron-Robert Zieler.
"Of course you are always looking for talent but what we have is one of the best number one keepers in the Premier League in Pepe Reina and then Brad Jones, who was a number three or number four keeper here but has really kicked on since the summer," said Rodgers at the time.
"In the games Brad has played he's showed he is a very loyal and supportive number two.
"Our job then below that is to ensure we have top young talents who can then hopefully challenge at the top end."
Despite his up and downs at Anfield, Jones hopes to be part of a successful side over the coming years.
"Obviously there was a difficult time in the middle of that, and three managers in that space of time is difficult because football is very much about opinions," he told liverpoolfc.com.
"Some people rate you more than others and I think I've found that in my time here at Liverpool - some managers have thought more of me than others.
"I said when I signed here it was a dream come true. 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."
In signing his new contract Jones follows the likes of Luis Suarez, Martin Skrtel, Daniel Agger and Jonjo Shelvey who have all extended their deals since Rodgers' arrival in the summer.