Keagan Dolly scored in the first half of extra time following a goalless 90 minutes at Makhulong Stadium.
Sundowns joined Polokwane City, Mpumalanga Black Aces, Baroka FC, Mbombela United, Golden Arrows, University of Pretoria, Free State Stars, Orlando Pirates, Bidvest Wits, Platinum Stars, Santos, Maritzburg United and Magesi FC in the next round of the tournament.
SuperSport United will visit Moroka Swallows, while Jomo Cosmos will host AmaZulu in the remaining last-32 ties on Tuesday.
The first half was dominated by the Brazilians, but their decision making was poor in the final third.
Downs had an early shot on goal in the eighth minute, but Khama Billiat fired wide of goal from the edge of the box.
In the 13th minute Themba Zwane should have given Sundowns the lead when he worked himself into space 16-yards out, but his shot on goal was weak and the keeper made a simple stop.
Highlands Park offered very little in attack in the opening half and never really threatened the Brazilians’ goal.
The NFD side started well in the second half and almost took the lead seven minutes after the restart when Simphiwe Tshablala broke through on goal, but he was denied in a one-on-one with Denis Onyango.
In the 62nd minute some great defending by Highlands’ Rudi Isaacs saw him clear a Katlego Mashego shot off the goalline.
Neither side could find a goal before the end of regulation time.
Five minutes into extra time Dolly broke the deadlock with a wonderful 24-yard shot that found the back of the net.
Highlands saw most of the ball after going behind, but could not find the equaliser.
Highlands Park XI: Kapini, Kalake, Isaacs, Christian, Ndlovu, Matloga, Kawela, Mvalo, Tshabalala, Verenga, Shalulie.
Sundowns XI: Onyango, Arendse, Soumahoro, Nthethe, Mbekile, Dolly, Mashaba, T. Zwane, Modise, Billiat, Ngele.