The 25-year-old Chile international completed his medical in London on Thursday, and will join up with his new team-mates at the end of July after spending some time on holiday following the World Cup.
Sanchez, who won the 2013 Primera Division title and 2012 Copa del Rey with Barca, is likely to make his first appearance for his new club at the Emirates Cup over the weekend of August 2-3.
The Gunners are also expected to bring in France defender Mathieu Debuchy from Newcastle, as manager Arsene Wenger looks to build on last season's FA Cup final success.
Bringing in Sanchez, who also had interest from Liverpool and Juventus, will present something of a personal coup for Wenger, who is understood to have spoken to the player face to face while working for French television at the World Cup in Brazil, convincing him Arsenal was the right place to further his career.
"As many people have seen during the World Cup this summer, Alexis is a fantastic footballer and we are delighted that he is joining us at Arsenal Football Club," Wenger told www.arsenal.com.
"Alexis will add power, creativity and much quality to our squad and we are all looking forward to him joining up with us in a few weeks.
"He has consistently produced top-quality performances at the highest level for a number of seasons now and we are all excited to see him integrate into the Arsenal squad. I'm sure all Arsenal supporters will join me in welcoming Alexis to our club."
Sanchez scored 39 goals over three seasons with Barca, having joined the Catalan side from Udinese.
He has netted 24 goals in 71 appearances for Chile and helped his country reach the last 16 of this summer's World Cup, where they were knocked out on penalties by hosts Brazil.
"I'm so happy to be joining Arsenal - a club which has a great manager, a fantastic squad of players, huge support around the world and a great stadium in London," the player said.
"I'm looking forward to meeting my new team-mates and to be playing for Arsenal in the Premier League and Champions League. I will give my best to Arsenal and want to make all the supporters happy."
The Gunners continue to work on bringing in other reinforcements to mount a long-overdue sustained assault on the Barclays Premier League title, having been clear at the top in January only to fade away and finish fourth.
While Debuchy will represent a Premier League-ready replacement for Bacary Sagna, who left on a free transfer to Manchester City, Wenger also has his mind on further additions.
QPR forward Loic Remy, rated at around £8million, is a potential target, having spent last season on loan at Newcastle, along with Bayer Leverkusen midfielder Lars Bender.
Reports have also linked Arsenal with Real Madrid's Germany midfielder Sami Khedira, although any possible move could well hinge on the personal terms.
Arsenal also need a goalkeeper, with number two Lukasz Fabianski having joined Swansea.
Midfielder Mikel Arteta has no intention of leaving the Gunners, according to his agent, with reports linking the Spaniard with Fiorentina.
Wenger will return to Arsenal after the World Cup final, ahead of a trip to New York at the end of the month.
After the Emirates Cup, Arsenal take on Premier League champions Manchester City at Wembley in the Community Shield.
Barca later confirmed the deal on their official website, and wished the player well.
"FC Barcelona and Arsenal FC have reached an agreement for the transfer of player Alexis Sanchez to the Premier League club," their statement read.
"FC Barcelona thanks Alexis Sanchez for his professionalism and dedication for the three seasons that he has been in the first-team squad, and wishes him all the best for the future."