Sky Sports News understands that Liverpool have paid £67m for the 25-year-old, who has signed a “long-term” contract at Anfield.
It surpasses the £34.7m Manchester City paid in 2017 to sign Ederson, Alisson’s understudy for Brazil.
“I’m really happy, it’s a dream come true to wear such a prestigious shirt for a club of this size that is used to always winning,” Alisson told Liverpool’s club website after completing the deal.
“In terms of my life and my career, it’s a huge step for me being part of this club and this family.
“You can be certain that I’ll give my all.”
Alisson also revealed his experience of playing at Anfield last season – when he was in goal for Roma’s 5-2 defeat by Liverpool in the Champions League – was a factor in his decision to move to the club.
“That game definitely did have an influence but I also watched other games,” he said. “I’ve seen other games where I could see the way Klopp had the team playing.
“That has made me really excited to be part of this group. Apart from the fact that the club is massive, the players that have come through here and the titles won.
“Hopefully I can play a part in the history being created here, win titles and be a part of the project that has been developed here in the past few years so we can see Liverpool rise once again.”
Alisson returned to Rome from John Lennon Airport in a private plane on Thursday evening after passing his medical.
Roma’s sporting director Monchi said the Italian club were reluctant to part with Alisson but that the offer from Liverpool was too good to turn down.
“When a very substantial offer comes in you have to consider it,” he said. “We weighed up the pros and cons and decided to speak to Liverpool.”
Roma tweeted their best wishes to Alisson on Thursday evening, saying: “Thank you Alisson and good luck with your new adventure.”
Alisson’s move to Liverpool had appeared unlikely earlier in the summer after the Premier League club were quoted an £80m asking price, Sky Sports News understands.
Liverpool manager Jurgen Klopp had accepted previous first-choice goalkeeper Loris Karius was going to be regularly scrutinised after he made a mistake in the 3-2 pre-season win over Tranmere.
Karius was at fault for two of Real Madrid’s three goals in May’s Champions League final defeat – after which it was revealed by US doctors the player was suffering from concussion.