The Chile international, who was also a reported target for Liverpool and Juventus, will add firepower to manager Arsene Wenger's ranks, having scored 19 goals in 34 La Liga appearances last season.
Sanchez has joined for an undisclosed fee, with Barca expected to sign Liverpool star Luis Suarez in the coming days to replace the Chilean.
Wenger was delighted to have sealed the attacker's signature and expects him to provide the London club with a significant boost.
He 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 at Arsenal Football Club.
"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 himself added: "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.
"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."
Sanchez, who won the Spanish title with Barca in 2013, scored twice at the FIFA World Cup as Chile's campaign was ended by hosts Brazil in the second round.
He will be tasked with helping Arsenal to secure their first Premier League crown since 2004, though the club did end a nine-year major-trophy drought last season after beating Hull City 3-2 in the FA Cup final.
