Dortmund announced on Monday that they have agreed a deal to sign the 24-year-old midfielder for an undisclosed fee.
The Slovenia international will sign a contract that will see him remain at Signal Iduna Park until 2019.
Kampl has scored 29 in 109 appearances for Salzburg in all competitions and won 14 caps for his country.
Rangnick has wished the German-born Kampl well as he prepares for his next challenge and feels he deserves his move to Bundesliga strugglers Dortmund
He told the Austrian club's official website: "For Kevin Kampl a move to the German Bundesliga is the logical next career step.
"He is joining a club that have regularly been right at the top in Germany and Europe in recent years.
"Kevin is an extraordinary player with a wide range of abilities who has developed fantastically with us over the past two-and-a-half years. We wish him all the best, and we hope he is successful in his new challenge."
