Zinchenko is the second signing Arsenal have made from the Premier League champions this summer after Brazilian striker Gabriel Jesus.
The deal can be boosted by a further £2m with performance-related add-ons.
Zinchenko has agreed a four-year contract at the Emirates Stadium.
The 25-year-old spent six seasons at Pep Guardiola's side and won four Premier League titles.
Zinchenko worked under Arsenal manager Mikel Arteta during the Spaniard's time as Guardiola's assistant at Man City.
Zinchenko is adept at both playing in midfield and at left back.
Following the arrival of Zinchenko, Arteta told the club's official website: "We're so happy Alex has joined us. He's a player that I personally know really well and continued to follow him after my time at Manchester City.
"Alex is a high quality player who will give us options and versatility. It's not only about the positions he can play but as well, the versatility he will give us in attack and defence.
"Alex is a person with great human qualities and character, and I'm delighted that everyone has made this huge effort to bring Alex to the club."
Arsenal's technical director Edu added: "I'm very pleased to see how we're working as a team here in the club. We put a plan together early and Oleksandr Zinchenko was a main focus on our list.
"We're pleased to have Alex with us now because he has the attributes which I'm sure are going to put our squad at a different level."
