The Spaniard, who has replaced Gerardo Martino at the helm, returns to the club where he spent eight seasons as a player and three years as the head coach of their B side.
"I'd like to thank [Josep Maria] Bartomeu, [Andoni] Zubizarreta and the whole board for its decision, for giving me their backing and for placing their faith in me and my team," Enrique told reporters at a press conference.
"Today is a very exciting, special day for me, on which we began to build a new Barca that will excite and get the results we're looking for.
"When I left Barca B, they said to me that it wasn't 'goodbye' but 'see you soon', and they've been as good as their word. Today is the realisation of everything a coach aspires to."
Enrique managed Celta Vigo in La Liga in the 2013-14 season.