The Bulls signed the Mali international, 20, from Salzburg, with whom they share the same owner, Red Bull.
In the summer of 2016, Keita was also signed by the Bundesliga side from their Austrian sister club and his performances caught the attention of top European clubs which saw him eventually land at Anfield during the summer.
And Rangnick believes the qualities of his new signing can see him follow in the footsteps of his Guinea counterpart.
"We are delighted to have signed Amadou," Rangnick told club website.
"There aren't too many players of his age who have so much potential. He has all the capabilities to follow in Naby Keita's footsteps.
"Of course, it's a shame that he won't be able to play for a bit due to injury, but we hope that he's back to full fitness soon, and will be able to play a part in the second half of the season."
BREAKING NEWS! ✍️#DieRotenBullen are delighted to announce the signing of Malian midfielder @doudouhaidara04 from @FCRBS_EN, on a five-year deal! 🔴⚪— RB Leipzig English (@RBLeipzig_EN) December 22, 2018
Welcome to Leipzig, Amadou 🇲🇱👋 pic.twitter.com/f4azeVl9Oa
Speaking on his five-year deal, Haidara said: "I'm very pleased to have joined RB Leipzig, and of course, get the opportunity to play in the Bundesliga. My priority now is to get back to full fitness as soon as possible.
"I'm on the right track but I still need time until I'm ready to play. I'm really motivated and can't wait to join up with the squad in Leipzig. Playing here will help me to further develop and also prove myself in the Bundesliga."