The 28-year-old Wales international will arrive in northern Italy on July 1 with his contract running until June 30, 2023.
As his contract will have ended, Arsenal will not be entitled to a transfer fee.
"As a result of the registration of the player, Juventus will incur additional costs of 3.7 million euros ($4.1 million), to be paid by July 10," the Serie A champions said.
The move brings to an end Ramsey's 11-year spell with the Gunners having moved from Cardiff City in 2008.
Over the past decade he has played 256 Premier League games, scoring 38 top-flight goals, and winning three FA Cup and two Community Shield.
He also had two loan spells at Nottingham Forest in 2010, and back at former club, Cardiff in 2011.
"I wanted to issue a personal statement for all the Arsenal fans who have been extremely loyal and supportive," Ramsey wrote on Instagram.
"You welcomed me as a teenager and have been there for me through all the highs and lows I have encountered during my time at the club.
"It is with a heavy heart that I leave after 11 incredible years in north London. Thank you.
"I will continue to give the team 100 per cent and hope to finish the season strongly, before heading on to my next chapter in Turin."
Ramsey has 58 caps for Wales, scoring 14 goals, and was a part of the Welsh side that reached the Euro 2016 semi-finals.
He becomes the third Welshman to play for Juventus following club legend John Charles and former European Golden Boot winner Ian Rush.
Ramsey had turned down the offer of a new deal at Arsenal, with clubs including Barcelona, Real Madrid, Paris Saint Germain, Inter Milan and Bayern Munich linked with the player.
"Everyone at Arsenal Football Club wishes Aaron Ramsey a happy and successful future," the London club said in a statement.
"Aaron's made a fantastic contribution to our club since joining from Cardiff City in 2008. He has always been a consummate professional and will always have a special place in the hearts and memories of Arsenal fans around the world."
Juventus have an 11-point lead on top of Serie A and are on track for an eighth consecutive league title.
They are through to the Champions League knock-out rounds and will play Atletico Madrid in the last 16 starting next week.
Ramsey will reportedly become Juventus' second-highest earner behind Cristiano Ronaldo earning 7 million euros ($7.8 million) a season.