The midfielder previously worked with the Northern Irishman during their time at Chelsea and says he was keen to link up with him again at the Merseyside club
"I know him from my Chelsea days when he was there and I can't see anything other than positive things," he told the club's official website.
"When I was hearing about Liverpool looking at certain managers, I was praying it was going to be Brendan - not just because he's not an unknown for me and I know his philosophy on football, but I think for the club the owners have made the right choice. He has just impressed everyone so much since he's been here."
Cole spent last season on loan at French club Lille after falling out of favor under Kenny Dalglish but made a significant impression in Ligue 1, scoring four times in 31 appearances.
The 30-year-old now looks set to stay at Anfield this term under Rodgers, and had nothing but praise for the new Reds boss.
"It's been a breath of fresh air. A new manager always brings a lift, but the gaffer has come in and really impressed everyone," said Cole.
"You talk to anyone - any of the players to the ground staff to the guys on the door - everyone is really impressed with him.
"The manager has made it clear, it's not just the players it's everyone in and around the club. For us to get the results we want, we've all got to come together."
The London-born midfielder also expressed his certainty that Rodgers would succeed on Merseyside, hailing his footballing philosophy.
"What you get from him is honesty, he's made that perfectly clear, and you're going to know where you stand," he added.
"I think a lot of clubs are steering towards that in English football now. He's going to be a success at Liverpool and if we all stick together and pull round, we can do some great things at this club."
Cole also insisted that the 39-year-old's possession-based football will benefit both the club and the English game, having played in a similar manner in France last season.
"That style is what I was brought up with at West Ham and Chelsea. At West Ham, we didn't have the top, top players, but we always tried to play," he continued.
"At Chelsea, we always tried to get the ball down and play and move it. For me, that is football. I love the style of play. At Lille, we played like that as well - getting the ball down and moving.
"I think sometimes in English football, people automatically think you need to be aggressive - but look at Spain. No-one works harder than Spain. They pass and move the ball quickly, and sometimes physically it's harder to do that because you're constantly on the move and you're fighting and have to be dogged."