Sahin passed a medical at the Reds' Melwood training ground before putting pen to paper on a move which will see him spend the rest of the current campaign at Anfield.
Arsenal were initially thought to be favourites to secure the services of Sahin and were in advanced negotiations, but interest from Arsene Wenger reportedly cooled to allow Liverpool to capture the Turkey international’s signature.
"I am happy to be here and I am looking forward to playing for Liverpool," he told the club's official website. "It makes me proud to have joined such a big club like Liverpool.
"After the first conversation we had, I was left impressed by Brendan Rodgers. He told me about how he wants to play football and what the goals are he wants to reach with Liverpool. I am looking forward to meeting the fans. I hope it will be a nice year with trophies."
Sahin revealed Real Madrid team-mate and former Liverpool star Xabi Alonso played a significant role in persuading him to make the switch to Anfield.
"When he heard Liverpool were interested in me, he started telling me about Liverpool," he added. "He is still in love with Liverpool, I think!
"He was saying, 'Go there, you will love it. The fans will take care of you and love you' and things like that. He said Anfield is the best stadium in the world. Xabi won a lot of trophies here, and hopefully we can do that too."
Sahin says he is relishing the prospect of playing for the Merseyside giants - and particularly captain Steven Gerrard, whom he greatly admires.
"Steven Gerrard is a huge name; not only in Liverpool but in world football," he continued. "I am looking forward to playing with him - I've played against him once.
"There are many other players that I could name - I could start from number one to 25. I know them all, not personally, but I know how they play football and what the style of Liverpool is.
"I am looking forward to meeting them all. If I have to choose one name, it would be Steven Gerrard - but it is not only Gerrard."
Former Dortmund ace Sahin will wear the No.4 shirt at Anfield.