Second half goals from Eleazar Rodgers and Teko Modise were enough for Johan Neesken's men to see out an AmaZulu side that played too conservatively on the day.
Anthony Laffor missed a penalty, and Carlington Nyandomo was sent off the hosts, after receiving two yellow cards.
Despite playing away from home, Sundowns took the initiative and came close to breaking the deadlock in the 16th minute, when both Tebogo Langerman and Laffor had attempts on goal deflected away by defenders.
The hosts created a rare chance in the 25th minute, but Nyiko Tshabangu fired wide, after being set up by Dulee Johnson.
A minute later Rodgers connected with a cross from Laffor, but Philani Zikalala watched the ball safely into his gloves.
In the 37th minute Sundowns were awarded a penalty for a handball by Nyadombo, but Laffor's low effort was saved by Zikalala, who dived the right way.
Sundowns took the lead in the second half, when Jabulani Shongwe played a defence-splitting pass to release Rodgers on the left, and the lanky striker placed his shot at the far post for his first goal for the Brazilians.
Johnson tried his luck with a free-kick just outside the box, and although he found the target his effort lacked the pace to cause problems for the goalkeeper.
AmaZulu were reduced to 10 men on 77 minutes, when Nyadombo received his marching orders for a second bookable offence.
Sundowns made use of their numerical advantage by spreading the ball wide, and in the 81st minute both Modise and Rodgers headed wide after connecting with crosses from Doctor Mampuru.
Modise made sure of the lead with a well-taken penalty, after Philani Cele was adjudged to have fouled substitute Nyasha Mushekwi in the box.