Usuthu overcame 10-man Polokwane City 2-1 in extra-time of their round of 32 encounter at the Old Peter Mokaba Stadium on Tuesday evening.
Taariq Fielies opened the scoring for AmaZulu in the 10th minute before Rise and Shine saw Monty Batlhabane pick up a straight red card for a rough tackle on Liam Bern.
Polokwane were back on level terms when Lebohang Nkaki converted a penalty before the half-time break.
With no further goals, the tie went to extra-time, where AmaZulu bagged a winner as Thandolwenkosi Ngwenya found the target.
Speaking after the game, Zwane dedicated the win to Zungu, on his birthday.
He said on FARPost: "We wanted to win this game for the president. He's been very supportive, and he's been there for us through thick and thin.
"We needed to make sure that we proceed to the next round, just to make him proud. He's always been proud of us whenever we do well as a team. So we must just enjoy what's left of today.
"Probably, we will give him more presents going forward, as we just want to see him happy. He's an amazing human being, and he has been trying to make the team grow for the KZN province," he added.
"So it's us that need to realise and fulfil or execute the plan that the club has for the future. This is one team that can help a lot of people in KZN. Young kids and many other players who are so talented. That's why we have to appreciate and embrace people like the president."
All the last 16 teams:
Orlando Pirates
Mamelodi Sundowns
AmaZulu
Stellenbosch FC
Golden Arrows
TS Galaxy
Sekhukhune United
Durban City
Richards Bay
AmaTuks
Casric Stars
Milford
Upington City
Vasco Da Gama
Minnows Mkhambathi FC
Jacksa Spears