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 club's managing director Ian Ayre said: "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."