The 32-year-old, who has 18 months left on his current deal, capped a dynamic performance with a clinical goal in Liverpool's 4-0 home win over Fulham at the weekend.
Rodgers, who took over from Kenny Dalglish at the start of the season, asserted that Gerrard remained a pivotal part of his plans at Anfield.
"I don't think there's any question [we want to extend his contract]," Rodgers told Liverpool's official website.
"It's vital. Steven has a real hunger to succeed. He's 32 years of age but he's still got so much left."
Liverpool's talismanic England midfielder had been linked with a move to Chelsea after the team's famous victory over AC Milan in the 2004-05 Champions League final in Istanbul.
Jose Mourinho, then Chelsea manager and now in charge of Real Madrid, remains an admirer, while even Manchester United boss Sir Alex Ferguson speaks highly of Gerrard.
"He applies himself every day, he eats well, rests well, recovers well, works well," said Rodgers, who has begun to get Liverpool playing more consistently in recent weeks.
"This is a guy who has led his life focused purely on being a footballer. And that allows you to go on playing well into your 30s."
Gerrard provided two assists on top of his sharp finish against Fulham and has played in every minute of Liverpool's 18 Premier League games this season.
"The run that he made for the goal, he's been doing that all season but the ball has just never arrived," Rodgers added.
"He's still been making the runs and his influence within the team has been outstanding.
"His influence for me, especially once he's started to understand what I'm looking for, has been absolutely first class."
Rodgers added: "[Against Fulham] he had everything in his game, and he's been an inspirational captain for us. I want him to stay beyond his current contract. There is absolutely no question about that."
