Rodgers revealed at a press conference earlier on Friday he was expecting the 22-year-old to sign a new contract soon.
And Kelly, currently out with a cruciate knee ligament injury, put pen to paper on a new deal at Anfield later in the afternoon.
The day after fellow defender Jamie Carragher announced his retirement at the end of the season, Rodgers was pleased to see another local lad commit his future to the club.
Rodgers said: "Martin has shown a commitment to our vision here. He is a young player who really impressed me when I came in.
"He was exceptionally good in the games he played earlier in the season."
The player, who injured his knee in the 2-1 home defeat against Manchester United in September, told his club's official website: "I'm delighted to sign. It means everything to me and my family.
"I've been at Liverpool since I was seven, so it's in my heart and I love the club.
"When you play in the Liverpool shirt, you have to show the fans what it means. I just love playing and trying my hardest for the supporters and for my family."
Kelly, who has made 54 appearances for Liverpool since making his debut in 2009, added: "Being a Liverpool supporter myself, it's brilliant to put the shirt on.
"I'm doing what I have loved doing since I was a kid. I'm over the moon.
"Today shows the faith that the club have in me and that the hard work I have put in since I was young has paid off.
"Hopefully I'll put the injuries behind me. We have excellent medical staff here, who are going to work me and keep me on top of my injury. I'll be coming back even stronger."
Liverpool managing director Ian Ayre added: "It's fantastic news and it runs on from various other players who have committed to us.
"I think everybody has been starting to see and enjoy Martin's potential and the work he has showed to get into the first team and these are the things we are looking for.
"He was very unfortunate to get his injury but he has worked very hard with the physios and the medical team and hopefully we'll see him back soon.
"Signing him up for the long term is very important for us."