Teko Modise and Man of the Match Themba Zwane got the goals for the dominant visitors, who joined neighbours SuperSport United, Ajax Cape Town and NFD side Vasco da Gama in the next round of the competition.
Sundowns started the tie brightly, as Zimbabwean playmaker Khama Billiat forced a save from the onrushing Jackson Mabokgwane in a one-on-one situation six minutes in.
The Premier Soccer League champions continued to dictate the pace of the game but struggled to break down a resolute AmaZayoni outfit.
Aces launched a rare attack midway through the first stanza as Onyekachi Okonkwo's dipping longe-range strike was unconvincingly dealt with by Ugandan goalkeeper Denis Onyango.
Soon after Downs forward Cuthbert Malajila beat Mabokgwane with a header off a Zwane cross from the right but the referee Robert Smith controversially disallowed the effort for an alleged foul on the keeper from Percy Tau.
Abia Nale should have done better 10 minutes before the break as Aubrey Ngoma teed him up nicely in the box. However, the former Kaizer
Chiefs attacker lifted his first-time attempt high and wide of Onyango's goal.
The Brazilians eventually broke the deadlock in the 58th minute, as Modise curled in a free-kick on the angle from the left, punishing
Mabokgwane for failing to guard his near post. It was 'The Navigator''s 10th goal in all competitions this season.
Billiat tried his luck four minutes later but his speculative shot was parried away by Mabokgwane.
However, Pitso Mosimane's side doubled their advantage in the 75th minute when Zwane clinically swept his effort home in the box.
The Tshwane outfit could have made it 3-0 late on but Mabokgwane did well to foil Billiat from close range.