The 48-year-old former Spain midfielder replaces Fernando Hierro, who stepped down as interim boss on Sunday.
Hierro – who is not returning to his role as sporting director – was placed in temporary charge of the team after Julen Lopetegui was sacked on the eve of the World Cup.
Enrique has been out of management since leaving the Nou Camp last June.
He takes charge of a team that were knocked out of the World Cup at the last-16 stage on penalties by hosts Russia.
Enrique, who played for Real Madrid and Barcelona, won the La Liga, Spanish Cup and Champions League treble as Barca manager in 2015.
The Spanish Football Federation (RFEF) announced his appointment on Monday and also named Jose Francisco Molina, a former Spain keeper, as their sporting director.
RFEF president Luis Rubiales said the body had been looking for “an undeniable leader in the dressing room, who sets a pattern and that no-one deviates from”.