The result sent Pirates top of the table on 41 points after 18 matches, three clear of Mamelodi Sundowns, who have a game in hand. Chiefs remained fourth on 30 points from 17 outings.
Pirates made their intentions clear from the first whistle. After early pressure and two corners inside the opening four minutes, the breakthrough arrived in the fifth minute. Moremi struck from outside the box, his left-footed effort flying into the bottom corner after Deon Hotto's delivery from a corner was only partially cleared.
Chiefs tried to settle, but Pirates controlled possession and tempo. Relebohile Mofokeng and Makhehlene Makhaula dictated play in midfield, while Bruce Bvuma was forced into saves from distance.
The second goal came in the 39th minute. Mofokeng slipped a precise through ball into the path of Appollis, who finished calmly into the bottom right corner to double the lead. Pirates nearly added a third before the break as Appollis fired just over during stoppage time.
Chiefs made changes at half-time, introducing Thabo Cele and Asanele Velebayi, but Pirates continued to create chances. Sipho Chaine denied Ethan Chislett just before the hour mark, while Bvuma kept out efforts from Moremi and Mofokeng at the other end.
Makgopa, introduced in the second half, sealed the result in the 78th minute. He rose at the near post to head home Moremi's cross from close range, putting the game beyond doubt.
Pirates finished with 15 attempts to Chiefs' five, underlining their dominance. Late efforts from Makabi Lilepo and Wandile Duba were comfortably dealt with by Chaine as the Buccaneers saw out a famous Soweto Derby win in style.