Midfielder Gerrard, the Merseyside outfit's 33-year-old captain who has spent his entire career with them, has put pen to paper on what is understood to be a two-year deal.
Giving his reaction, Rodgers told LFC TV: "I'm absolutely delighted, as you can imagine. I've talked throughout pre-season about wanting to bring other players into the group, but this was one of the - if not the - most important signings for us.
"He's an icon here at the football club - a real iconic figure for players and supporters. I think he's still got a good few years left yet. His level of performance last year was outstanding."
Aside from missing the last two games of the campaign due to a shoulder problem, Gerrard started every other one of Liverpool's Barclays Premier League matches last season - Rodgers' first in charge of the Reds, which saw them finish seventh.
And having signed his new deal, Gerrard - who has made 630 appearances for Liverpool, scoring 159 goals, and won the 2005 Champions League, the 2001 UEFA Cup, two FA Cups and three League Cups - has spoken of his excitement about the future.
The Reds academy product, a first-team debutant in 1998, said: "I've got a lot of confidence and belief in my team-mates and in the coaches we've got.
"That's the reason why I feel as if we can move forward together and get success back soon. I won't lie to you, I'm really excited at the moment."
Asked about his relationship with Rodgers, Gerrard - also the England skipper - said: "I think it's perfect. I've got a lot of respect for Brendan as a coach, but also as a guy as well.