Ashley du Preez netted before and after a Saleng goal to give Chiefs a 2-1 half-time lead following a fast tempo opening 45 minutes in 30 degrees Celsius (86 Fahrenheit) heat.
Thabiso Lebitso brought Pirates level a second time on 58 minutes and Saleng scored the decisive goal after 68 minutes as Chiefs slipped to seventh.
Matches between Pirates and Chiefs, whose rivalry began in 1970, draw the biggest attendances for a club match in Africa as Soccer City has the largest capacity.
But many clashes have been anti-climactic in recent seasons with caution preferred to creativity, leading to cagey, low-scoring affairs.
When Du Preez opened the scoring after only six minutes, however, Pirates opted to attack and this triggered a thrilling seesaw struggle for supremacy.
“It was a fantastic derby with every player exerting incredible energy,” said Jose Riveiro, the Spanish coach of former African champions Pirates.
“This match proved that we can compete against any team on our day. What we have to do now is close the gap between our best and worst performances.”
Chiefs caretaker coach Cavin Johnson said: “What a match. Both sides had moments of brilliance and the crowd loved it.
“I am disappointed because at one stage we had all three points only to end with none. I am going to bite my tongue and calmly tell the players where they went wrong.”
Several hours after the derby, Sundowns maintained a relentless match toward a seventh straight title with a 2-0 home win over bottom-half Chippa United in Pretoria.
An early thunderbolt from captain Teboho Mokoena and a late goal from substitute Tashreeq Matthews delivered maximum points.
Third-placed Stellenbosch stretched an unbeaten run in all competitions to 16 matches as Iqraam Rayners equalised from the penalty spot in a 1-1 draw with visiting Cape Town Spurs.
© Agence France-Presse
Photo: Twitter @orlandopirates