The Englishman joins Villa following an unsuccessful stint at Old Trafford, where he made 83 appearances and registered 12 goals and six assists.
Sancho spent last season on loan at Chelsea, where he made 15 goal contributions in 41 games, winning the UEFA Conference League in the process.
"Aston Villa is delighted to announce the signing of Jadon Sancho from Manchester United," the club said in a statement.
"Sancho has been playing at the highest level of the club game since he was a teenager, particularly impressing for Borussia Dortmund during his time in the Bundesliga.
"Last term the winger was on loan at Chelsea, where he helped the London club win the UEFA Conference League, scoring in the final against Real Betis."
Villa's acquisition of Sancho represents an important addition to their attack - an area they had been trying to strengthen all window.