The club has not confirmed the length of the contract but it is understood it runs until June 2020.
Coutinho, an ??8.5m signing from Inter Milan two years ago, has become a key player for Liverpool this season.
He told the club’s website: “I’m really happy. It’s a dream come true playing for this great club and being part of this squad.
“Today I’ve committed my future to the club and for this I’m feeling very pleased.
“This club is great and the fans have always supported me, so I’ve been eager to sign this new deal since conversations started. Today is a very happy day.
“It means so much. This club has given me the opportunity to play and believed in me even though I hadn’t been playing regularly at my previous club.
“I’ve had a great reception since my first day here. As I always say, Liverpool is a big family and I’m a happy person here.”
The Brazilian provided both assists in Saturday???s 2-0 win over West Ham, leading manager Brendan Rodgers to brand the 22-year-old “a little magician”.
???Young Phil is performing at a real high level and it???s a great to see because he???s a little magician with the ball,??? Rodgers said. ???The rhythm of his game is good. His courage and bravery to get on the ball is fantastic.
???If he can continue to improve ??? he can improve his goal-scoring ability ??? then he will get into that bracket of being world class.
???He has got a wonderful strike on him, inside and outside the box. It will just be about arriving in areas where he can make movements without the ball because once he is in there he has got a decent finish on him.
???His game has been about making goals but I think as he gets more experience he will understand better and there will be other players to make those final passes. That will allow him to penetrate a little bit further forward.
???He will be looking to improve the accuracy in his shooting. Sometimes you see him dragging the shot, but at the minute I don???t have many complaints about him.
???He is pressing the game really well, he is aggressive and hunting the ball. You see his quality in his two assists at the weekend. They were sublime.
???The first pass was incredible invention, he was half looking away when he made the pass and the second one looks an easy pass but he has put it to the correct side for Daniel???s touch and finish. Hopefully, in the years to come, the goals will come more readily for him.???
Rodgers has compared Coutinho???s role to that of Real Madrid pair Luka Modric and Toni Kroos and says his lack of goals – just two this season – is not an issue.
???You look at Modric, he doesn???t have a big goal scoring record but he is a world class player,??? Rodgers added..
???He is a continuity player. He takes the ball in the middle of the field, he can make the passes, serving the ball from behind and he gets the odd goal from outside the box.
???He is a top-class player who opens the door for the team in the centre of midfield. Toni Kroos is another of that type – a wonderful footballer and great passer.
???Those two maybe don???t get as many goals as their technique would suggest but that goes back to their role in the team. Their role is to sit behind the ball and serve the ball and create the game for others – not necessarily to score 15 or 20 goals a season.
???Phil???s numbers will improve but it???s not the be all and end all if he doesn???t get 20 goals. His functions are elsewhere in the team.???