Swede Backe's contract was not renewed after the Red Bulls were defeated in their Eastern Conference semi-final against DC United on Thursday.
Roxburgh was manager of Scotland from 1986 to 1993 and was then appointed UEFA's first technical director where he was charged with developing coaching across the continent, a position he held until recently.
Roxburgh will link up with Frenchman and former Liverpool manager Gerard Houllier, who works for the Red Bull company as their global director of football also covering teams in Salzburg, Leipzig and Sao Paulo, Brazil.
"I am thrilled to have Andy on board," Houllier told the team's website.
"To have someone of Andy's calibre and experience shows the strength of our commitment to make the New York Red Bulls one of the best franchises in US soccer," he added.
Roxburgh, 69, will be responsible for all football activities at the club and said he was looking forward to working in the North American league.
"I'm delighted to be getting involved in Major League Soccer which has an impressive reputation as a fast growing and highly competitive league," he said.
Backe, who had previously worked as Sven-Goran Eriksson's assistant at Manchester City and with the Mexican national team, had been coach of the Red Bulls since 2010.
Roxburgh and Houllier will now turn to the appointment of the next Red Bulls coach. Local website Soccer By Ives has reported that Houllier's former Aston Villa assistant and former Scotland international Gary McAllister would be under strong consideration.