The Austrian energy drink company said the 59-year-old German would take charge of their global portfolio from October.
Schmidt has previously coached Red Bull Salzburg, Bayer Leverkusen, Beijing Guoan, PSV Eindhoven and Benfica.
Former Liverpool and Borussia Dortmund boss Klopp, also 59, was appointed Germany coach on July 24.
Red Bull owns clubs in Germany, the United States, Brazil and Austria including RB Leipzig, Red Bull Salzburg and New York Red Bulls. The company also has a stake in Ligue 1 French side Paris FC.