Downs survived the earlier pressure after countless shots from within and outside the box, with Denis Onyango called into the action on numerous occasions.
The hosts grew stronger as the half went on and created a few mouthwatering openings, which Hlompho Kekana, Gaston Sirino and Percy Tau failed to convert.
Kekana got goal-side on the edge of the box and fired a shot that came off the outside post, while Sirino lined-up for a volley later on from Themba Zwane’s cross but blazed well over.
Tau, meanwhile, broke free on the left before cutting inside for a curler but was unable to hit the net let alone find the target.
Sundowns eventually went in front after Wayne Arendse broke into the box from a corner to turn the ball home with his thigh on 34 minutes.
The visitors looked for a quick response but the wet conditions made it difficult for them to display their natural offensive game as they went into the break trailing.
In the second half, the Burundians continued searching for a way back as Hussen burst through the box, but was stopped on his way by Arendse and Onyango.
The 2016 African champions looked dangerous every time they got on the ball and came close to a second goal after 51 minutes, but Tau was denied from close range.
The second goal did come after good work from Sirino, who rattled the crossbar before Sibusiso Vilakazi finished off the move on the rebound for his first goal the tournament on 55 minutes with a flying header.
The Uruguayan fed another sublime delivery into the path of Tau who managed to bring the best out of the Burundian shot-stopper.
The away outfit were reduced to 10 men late on after Caleb Bimenyamina saw red for a two-footed challenge on Tebogo Langerman in added time, but by then, Sundowns were home and dry as the won the tie 2-0 on aggregate.
Sundowns: Onyango, Morena,Nascimento, Arendse,Langerman, Mabunda,Kekana, Zwane, Sirino (Manyisa 83’),Vilakazi (Ngoma 81’),Tau (Brockie 89’).
Subs: Brimah, Brockie, Manyisa, Madisha, Laffor, Zulu, Ngoma.
Rayon: Ndayishimiye, Rutanga, Manzi, Mutfinzi, Mukunzi, Mbondi (Ismaila 77’), Hussen, Usengimana, Nyandwi (Bimenyimana 66’). Kwizera, Mugisha (Muhire 62’).
Subs: Ndacyayisenga, Mugabo, Bimenyimana, Muhire, Irambona, Ismaila, Niyonzima.