The 19-year-old's deal with the Reds expires next month and though an offer was on the table, with the player's representatives unable to broker a deal with the club.
As such, the Barclays Premier League outfit have decided to release Rossiter early from his contract - which was due to expire on June 30 - and he has put pen-to-paper on a four-year £10,000-a-week deal at Ibrox Stadium.
Reds to receive nominal fee for Rossiter's development
With the Scouser making a cross-border transfer, Liverpool will not be forced to go to an independent tribunal over any potential fee. Instead, they will be owed just £250,000 for the England youth international once the deal is formally announced in July, when the transfer window officially opens.
The Everton Valley-born teenager, who made five first-team appearances for the Reds this season including his Premier League debut back in August, has been at the club since the age of six.
But injuries have wrecked his season, with a hamstring problem keeping him out of action since last year - meaning he has not appeared since a 0-0 Europa League group stage draw with FC Sion back in December.
With several clubs keeping a watchful eye over Rossiter's situation, it was the Scottish outfit who eventually swooped to get a deal, with their head of recruitment Frank McParland, formerly the Academy director at Liverpool, convincing manager Mark Warbuton that the tough-tackling midfielder was worth signing.
Rangers view Rossiter's capture as one that can potentially catapult them into contention for the Scottish Premier League title in their first season back in the top-flight, with the young midfielder having once been highly-rated on Merseyside.
He made his debut for the Reds' Under-18 side on his 15th birthday and immediately becomes one of the Gers' highest-paid players, just a few thousand pounds per week behind skipper Lee Wallace as they look to challenge Old Firm rivals Celtic next term.
Rossiter delighted to make move to historic club
Speaking to Rangers' official club website, Rossiter described himself as "made up" to be at a "great club with a lot of history" and said he's "really looking forward to getting started."
He also called it "a great feeling" and "an honour" to go from "a massive club like Liverpool to another massive club like Rangers" and referred to his prior experience of working with McParland, adding: "I am looking forward to working with him again and getting started here."
Rossiter admitted that Rangers "didn't need much selling" and that he chose them "because of the history it [the club] has got" and the fact that it's "a massive club."
He continued: "Rangers is a club which wins trophies and winning trophies would be brilliant," when asked about his hopes and aspirations in Scotland. "They have two already and are going for a third next week, so it is a club that is always going to win things and next season, hopefully, I can help with that."
The Reds have also let go of 20-year-old defender Dan Cleary. The Republic of Ireland youth international, who was part of the senior squad which travelled to the Far East and Australia last summer for pre-season, only made the first-team squad once under former boss Brendan Rodgers.
As such, he has been released after being told he wouldn't feature under Jürgen Klopp. He took to Twitter to say: "Been a tough day saying goodbye to everybody in the club. I’d like to say thanks to everyone at LFC for everything. Time to move on for me."