Backe won four Danish titles in charge of FC Copenhagen and Aalborg and most recently was assistant to compatriot and former England manager Sven-Goran Eriksson with Manchester City, Mexico and English League Two side Notts County.
New York will have a distinctly Scandinavian feel when they kick off in March at their newly-built stadium in New Jersey with 57-year-old Backe working alongside general manager and sporting director Erik Soler who is Norwegian.
Red Bull assistant coach Richie Williams and goalkeeping coach Des McAleenan will continue in their roles.
"I have followed Hans's career for many years and I think he will be an excellent coach for this team," said Soler.
"Hans brings a vast amount of experience from a variety of different clubs in Europe and he has been a fantastic mentor for many young and established players.
"We are confident that he will be able to help our club accomplish our goal of competing for an MLS Cup."
"I am pleased that Richie Williams and Des McAleenan will remain with the club," Soler added. "They are both important to the success of the organisation with their depth of player and league knowledge."
"I am extremely excited to join the Red Bulls," Backe said.
"It is evident with the stadium and the resources available that Red Bull is very committed to making this club the best in Major League Soccer. I hope that I can bring my expertise to the club and build a winning squad for our supporters."
The Austrian Red Bull energy drink company took over the club in 2006 when it was known as the New York/New Jersey MetroStars.
Last season the team failed to make the playoffs and head coach Juan Carlos Osorio quit before the end of the campaign with American Williams taking charge for the final games.
