The 25-year-old scored 47 goals in 141 appearances for Barca and netted two goals in the World Cup in Brazil.
He said: “I’m so happy to be joining a club which has a great manager, a fantastic squad of players and huge support around the world.
“I’m looking forward to playing in the Premier League and Champions League.”
The Gunners also remain on course to sign French right-back Mathieu Debuchy, 28, from Newcastle United.
Liverpool were keen on recruiting Sanchez as part of the deal that is likely to take striker Luis Suarez from Anfield to Nou Camp, but that is now expected to be a straight cash transfer.
Juventus had also shown a strong interest in Sanchez but the Italians are reportedly closing in on Real Madrid forward Alvaro Morata instead.
Arsenal manager Arsene Wenger said: “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.
“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.”
Sanchez, who has made 71 appearances for Chile, scoring 24 goals, helped his country reach the World Cup last 16, where they lost on penalties to Brazil.
He 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.
In addition to pursuing Debuchy, Arsenal is still in the market for a holding midfielder and back-up goalkeeper.
Their move for the defender is at an advanced stage but the clubs are yet to agree a fee, with Newcastle, who signed Debuchy from Lille for £5m in 2013, understood to be seeking in the region of £10m.
Debuchy would replace Bacary Sagna, who left Arsenal to join Manchester City.