Leipzig, founded in 2009 with backing from drinks firm Red Bull, will play in the group stage after a 4-1 win at Hertha Berlin on Saturday guaranteed a top-three finish in the Bundesliga.
“At the start of the season, we just talked about being worry-free [from relegation],” Rangnick said.
“I am simply proud.”
“When we play in the Champions League, we’ll have to strengthen the squad,” added 58-year-old Rangnick, who has previously managed German clubs Hoffenheim, Schalke and Hannover.
“Whether three or four new additions will be enough, I do not know. Perhaps we’ll need five or six and we need more quality across the board.”
The win over Hertha Berlin was Leipzig’s 20th in 32 league games since winning promotion last season.
German forwards Timo Werner and Davie Selke scored two goals each.
Twenty-year-old midfielder Oliver Burke, who became the most expensive Scottish player when he joined Leipzig for a fee of around £13m in August 2016, was an unused substitute.
