Financial details were not disclosed but Sky Sports said Arsenal would pay 30 million pounds ($36.05 million) up front with two million pounds in add-ons for the defender.
Zinchenko made more than 100 appearances in all competitions for City, including the 2021 Champions League final where Pep Guardiola's team finished runners-up to Chelsea.
"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," Arsenal manager Mikel Arteta said.
He becomes Arsenal's second recruit from City in the close season after Brazil striker Gabriel Jesus, and their fifth signing overall after winger Marquinhos, goalkeeper Matt Turner and attacking midfielder Fabio Vieira.