Rangnick, who will also be in charge of development at sister club RB Leipzig in Germany is joined by former France and Liverpool coach Gerard Houllier, who takes over as global football director for the company which owns Salzburg and several other clubs.
The cash-rich club, owned by the Red Bull energy drinks company, also appointed Roger Schmidt, former coach of German second division club Paderborn, as Salzburg's new head coach.
"Salzburg will tackle the new tasks and challenges with a new team," the club said in a statement, saying more details would be available at a news conference on Monday.
Salzburg have dominated Austrian football since the club was taken over by Red Bull in 2005, winning the league four times in the last six seasons.
They also won their first Austrian Cup in May.
Rangnick had also been in talks with West Bromwich Albion regarding their vacant manager's position before it was filled by Steve Clarke.