Enrique had been widely tipped to succeed Martino in recent weeks, and speculation that his appointment could be imminent increased when the 44-year-old resigned as manager of Celta Vigo on Friday.
Barcelona have now confirmed that he will take charge of the team next season as the club looks to bounce back from a disappointing campaign that saw them miss out on the league, the Copa del Rey and the Champions League.
The appointment sees Barca return to their policy of preferring managers with past experience of the club, as Enrique spent eight years as a player at Camp Nou as well as three years as boss of the team's 'B' side.
Enrique, who also played for bitter rivals Real Madrid during a distinguished career, made more than 200 league appearances for Barcelona and helped them to two league titles, two Copa del Reys and a Cup Winners' Cup.
Barcelona missed out on the league title this season having only managed a 1-1 draw with eventual champions Atletico Madrid on the final day.