Thiago's contract with the Bundesliga champions was due to expire next summer and he had been linked with a move away from the club for several months.
"I think it's an amazing feeling," Thiago told Liverpool's website. I was waiting for this moment for a long time and I am very, very happy to be here.
"When the years are passing, you are trying to win as much as you can -- and when you win, you want to win more. I think this club describes what I am as well; I want to achieve all of the goals, win as many trophies as possible.
"It also [has] this kind of family [feeling] that I need because we [like to] have a very close relationship with the club and I think I will feel this, I will have this feeling, with this club as well."
Liverpool manager Jurgen Klopp had been determined to add the former Barcelona player to his squad earlier this summer, but financial restrictions at the club in the wake of the coronavirus pandemic had left the Anfield hierarchy reluctant to sanction an expensive move for a 29-year-old.
Klopp had been told that a first-team player would have to leave the Premier League champions in order to create space on the wage bill for Thiago.
However, sources told ESPN that Klopp was given the green light to secure Thiago last week, without the need for the club to sell first.
Klopp said of his new signing: "He is an absolutely exceptional player. It's difficult to improve a team like us at the moment in a normal way.
"It's got nothing to do with other teams but if we had £200-300 million we could find some players and even then, it's not easy.
"In this specific case, he's a player who really brings in a different dimension to our game. He has some skills which are rare but they suit us pretty well. It says nothing about the other players, it's only different and that's what I was very interested in.
"Apart from that, he was pretty excited about the opportunity as well so there was not a lot of convincing needed. That helps as well if somebody really wants to be part of a specific project that's nice and everything came together. It's a win-win situation and I'm really happy that we could do it."
Thiago has been an integral part of the Bayern side that has dominated German football in recent years.
He joined Bayern in 2013 from Barcelona, where he won two La Liga titles, and has since won seven Bundesliga titles.
Last season he helped the team to a remarkable Treble where they topped the Bundesliga and also won the German Cup and Champions League.
"I still clearly remember how Pep Guardiola said at the beginning of his tenure in Munich, 'Thiago or nothing.' In hindsight, I'm very happy that we fulfilled Pep's demand at the time," Bayern CEO Karl-Heinz Rummenigge said."Thiago has developed from a talented 22-year-old into a true leader, who we hugely appreciate above all as a person.
"He wanted a new sporting challenge in another league. We've granted that request and agreed a transfer with Liverpool. On behalf of all of FC Bayern, I wish Thiago and his family all the best for the future. A truly distinguished player like him will always be a friend of our club."
Sporting director Hasan Salihamidzic added: "I believe we all regret that Thiago now wants something new in his life as he approaches 30. He's an extraordinary footballer and person. I'm sure that he'll remain close to FC Bayern.
"With us, he became one of the best players in Europe and won everything there is to win at club level with FC Bayern. His children were born in Munich and his family have made their home in Munich. He won't forget that. I wish him good health and hope he finds what he's looking for."