Prior to that he had been assistant manager to Eriksson, the Notts County director of football, at Manchester City and with the Mexico national team.
Backe will replace another former Manchester City assistant coach, Colombian Juan Carlos Osorio, who was a member of Kevin Keegan's staff at Eastlands.
Osorio resigned midway through the 2009 season as the Red Bulls, MLS Cup finalists in 2008, went on to finish with the worst record in the league, winning just five games out of 30.
The Red Bulls will move from the soon-to-be-demolished Giants Stadium to a new soccer-specific 25,000-seater ground in nearby Harrison, New Jersey, for the 2010 season and Backe said that was one of the attractions for making his move to MLS.
"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."
Backe has won four league titles in Denmark as head coach at FC Copenhagen and has also coached Panathinaikos and SV Salzburg, now also owned by drinks manufacturer Red Bull.
"I have followed Hans' career for many years and I think he will be an excellent coach for this team," New York Red Bulls general manager and sporting director Erik Soler said.
"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 a MLS Cup."