The 44-year-old has put pen to paper on a three-year contract.
It will see him replace Domenec Torrent, who was sacked last November.
Deila was successful in Scotland, winning the League Cup and two domestic league titles with Celtic.
He has also coached Valerenga in Norway.
"When the opportunity came up to join New York City FC as their new head coach, it was a dream come true for me to manage an amazing club in a great city," Deila told the club's website.
"I'm ready to see our supporters and hopefully we'll have a wonderful relationship. I'm going to do everything I can to play good football the City way and I'll give everything from myself to the club and I'm sure we're going to achieve fantastic things in the future."