The 25-year-old has joined on a long-term contract for a fee undisclosed by the club, but Sky Sports understands the figure to be in the region of £35million.
Sanchez, who starred as Chile reached the last-16 of the World Cup, said: "I'm so happy to be joining Arsenal - a club with a great manager, a fantastic squad of players, huge support around the world and a great stadium in London."
Considered one of the most exciting attacking players in world football, Sanchez made over 100 appearances in all competitions for Barcelona after a 2011 move from Udinese, scoring 43 goals in three seasons with the Catalonians.
But despite his contribution to their 2013 La Liga and 2012 Copa del Rey triumphs, among a number of other domestic trophies, Barcelona believed he was no longer required at Camp Nou beyond this summer.
Sanchez was courted by a number of clubs across Europe, with Juventus and Liverpool showing the strongest interest alongside Arsenal, but the player made his desire to move to north London clear to his representatives.
Sky sources learned he had arrived at Arsenal's London Colney training complex to undergo a medical on Thursday and upon completing the deal, manager Arsene Wenger said: "Alexis will add power, creativity and much quality to our squad.
"He has consistently produced top-quality performances at the highest level and we are all excited to see him integrate into the Arsenal squad."
Barcelona, meanwhile, said: "We thank Alexis Sanchez for his professionalism and dedication and wish him all the best for the future."
Sanchez began his career at Chilean side Cobreloa before moving to Colo-Colo, one of the biggest clubs in his home country, where he twice won the Primera championship in 2006 and 2007.
His prowess earned him a move to Argentine giants River Plate, where he won the league in his first and only season, before moving to Italy with Udinese in 2008 where he scored 20 times in 95 appearances.
Sanchez made his international debut in 2006 and has gone on to score 24 goals in 71 caps. He was integral to their success in Brazil this year, scoring twice and providing an assist as Chile reached the last-16 for only the fourth time in their history.
The player will now take leave for holiday before joining up with his new meamates later in July, and the club expect Sanchez to make his first Arsenal appearance at the Emirates Cup on the weekend of August 2.
Sanchez is Arsenal's first signing of the summer but they are also understood to be close to the signing of France right-back Mathieu Debuchy from Newcastle.
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