Rose was shown the door after defeat to Borussia Monchengladbach on Saturday.
Former Liverpool and Borussia Dortmund coach Jurgen Klopp is now Red Bull's Global football director. It's believed he was consulted about the decision.
RBL are only three points from the top four positions and play in the semi-finals of the German Cup on Wednesday against Stuttgart.
Leipzig's managing director for sport, Marcel Schafer, said: "We have believed in our co-operation with Marco and his team for a very long time and, up to now, tried everything to turn things around together.
"Given the development and the continued lack of results, we're all firmly convinced that we need new impetus for the remainder of the campaign in order to achieve our aims for the season."