The South Africans replaced DR Congo's AS Vita Club, who were disqualified for fielding an ineligible player.
Sundowns will now take on Enyimba, Zamalek and Entente Setif. Group A consists of Zesco, Al Ahly, Asec Mimosa and Wydad Casablanca.
The top two teams in each group will play in the semi-finals.
The teams will play each other home and away in the group stage, with the first round of matches scheduled for the weekend of the 17 June.
Also on Tuesday the daw for the Confederation Cup was made, with title favourites TP Mazembe of DR Congo and Tunisia's Etoile Sahel kept apart in the two groups of four.
Mazembe, who have won nine Confederation of African Football competitions, were placed in Group A with Mouloudia Bejaia of Algeria, Young Africans of Tanzania and Medeama of Ghana.
Defending champions Etoile are in an all-North African Group B with FUS Rabat and Kawkab Marrakech of Morocco and Al Ahly Tripoli of Libya.
As in the Champions League, the group winners and runners-up qualify for the semi-finals.
Mamelodi Sundowns have been drawn in Champions League Group B, only hours after they were reinstated into the competition at the last-eight stage.