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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 the 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 are 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.

He was treated in hospital for injuries to his arm, back and face after being bitten by the dogs in Buttershaw Drive, Bradford, at about 07.30 BST.
West Yorkshire Police said armed officers secured the dogs in a garden following reports of the attack.
Sgt Richard Sheard said two men have been questioned in connection with the incident.
He said both dogs were taken away by specialist officers along with a third dog, a red nosed pit bull.

The 20-year-old, who has three caps, spent five years at Fulham before leaving for the Cherries in the summer on a four-year contract.
He made his first Bournemouth start in Saturday's FA Cup defeat at Millwall.
Rangers are also in talks with Arsenal over the loan signing of attacking midfielder Jon Toral, who is currently on loan with Grenada in Spain.
"I think the two clubs have similar styles which makes it an easy transition and all the boys have helped me out on my first day," Hyndman told Rangers TV.
"The philosophy here is the same as Bournemouth in the way that they like to press but when they have the ball they like to keep it in attack.
"In that way it is very similar which will hopefully make my transition a lot easier.
"There were a lot of reasons I wanted to sign for Rangers, it is a massive club, they are fighting for Europe and there are good players here.
"I think it is a good opportunity for me to play in, not just a good environment, but in front of 50,000 fans at home which is absolutely unbelievable."
Meanwhile, talks are on-going with Arsenal over Spaniard Toral, 21, who has only featured six times for Granada this season and was an unused substitute in their 5-0 defeat away to Real Madrid on Saturday.
Toral played under Rangers boss Mark Warburton in 2014-15 when he was in charge at Brentford, and spent last season on loan at Birmingham.
Chile international striker Alexis Sanchez has signed for Arsenal from Barcelona on a long-term contract for a fee in the region of £35m.
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A man suffered serious injuries when he was attacked in the street by two American pit bulls, police said.
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Rangers have signed USA midfielder Emerson Hyndman on a six-month loan deal from Bournemouth.