The two sides drew 1-1 in the first leg of the tie with Walid El Karti netting an 94th minute equaliser after a 54th minute Lucas Ribeiro opener at Loftus Stadium.
Fiston Mayele opened the scoring in the second round tie and Ahmed Samy seemingly put the result out of reach for Downs, while Iqraam Rayners gave the Betway Premiership side a glimmer of hope with his goal after 75 minutes.
Sundowns have now been losing finalists twice, but for Pyramids this was their debut campaign in the CAF Champions League.
A delighted Jurcic said on IOL: "We achieved the accomplishment and were able to win the African Championship, and we deserved it.
"We truly deserved this title. This trophy is a gift from God for our efforts."
The Croatian coach added: "I want to thank the players for everything they have done. They truly deserve to be African champions.
"I thank the supporters for their presence and their unwavering support. They played a vital role in our victory.
"There were many fans who came behind the team. I asked the Egyptian fans to support us, and I thank them for coming and for giving us the energy that helped us win. We — supporters from different clubs — were all together."
Mamelodi Sundowns coach Miguel Cardoso admitted his players are suffering after their CAF Champions League final defeat to Pyramids, acknowledging that it is never nice to play the final and you don't win.
Mamelodi Sundowns saw their CAF Champions League dream slip away on Sunday night as they fell 2-1 to Egyptian side Pyramids FC, who lifted the prestigious continental trophy for the first time in their history.
Mamelodi Sundowns head coach Miguel Cardoso is on the cusp of etching his name into African football history as he prepares his side for the decisive second leg of the CAF Champions League final against Egypt's Pyramids FC this Sunday in Cairo.
Mamelodi Sundowns will need to defy the odds in Cairo on Sunday night as they face Pyramids FC in the second leg of the CAF Champions League final, with everything on the line at the 30 June Stadium.
