The Reds will pay the Glasgow giants an initial £2m for the 18-year-old with a further £3m to come depending on appearances and other add-on clauses.
Rangers have been forced to sell the highly rated youngster, who was just entering his first year of his contract, after he rejected a new offer to stay at Ibrox.
Wilson said: “There were other clubs interested in me who are in the Champions League, but once Liverpool came in there was only one team I wanted to go to.
“This is a massive challenge for me. This is one of the best leagues in the world and I will be playing for one of the biggest teams in the world.
“Coming here from Scotland, people may say I might not be ready for this, but that is a challenge I want to put on myself and that’s the reason I came here.”
Wilson is the third arrival at Anfield during the close season, following the free-transfer signings of Joe Cole from Chelsea and Milan Jovanovic from Standard Liege.
His 24 appearances for Rangers last term earned him the Scottish Football Association and Football Writers’ Young Player of the Year awards.
Javi Gracia said it was always going to be an uphill battle for Watford after conceding so early against Liverpool on Saturday.
Schalke 04 boss Dominico Tedesco paid tribute to goalkeeper Ralf Fahrmann after his 76th-minute penalty save proved crucial in a 1-0 win over Wolfsburg.
Mattia De Sciglio stunned that Juventus were shocked to be held to a goalless draw with SPAL on Saturday.
Sunderland have suspended midfielder Darron Gibson with immediate effect after he was charged with drink driving.
Jose Mourinho's recent bad frame of mind was still present despite Manchester United beating Brighton 2-0 at Old Trafford on Saturday, a result that puts them into the last four of the FA Cup.
Mohamed Salah continued his astounding goalscoring form on Saturday, hitting four goals as Liverpool thrashed Watford 5-0 at Anfield.