"We are delighted with Alejandro. He is so talented, young and he showed us his wish to help our project. We are really happy," Unai Emery said in a statement.
Garnacho, 22, arrives at Villa after a difficult season at Chelsea, where he struggled to secure a regular starting role following his £40 million move from Manchester United last year. He made 43 appearances in all competitions, scoring eight goals.
The winger left United last summer after spending five years at Old Trafford. He scored 26 goals in 144 appearances for the club, but had fallen out of favour under former head coach Ruben Amorim before his departure.
A product of Atletico Madrid's academy, Garnacho was included in Argentina's 55-man preliminary squad for the World Cup but did not make the final cut.