Goals from Yamela Mbuthuma (13'), Oswin Appollis (45') and Relebohile Mofokeng (58') sealed a comprehensive victory at the Orlando Amstel Arena.
Usuthu are chasing a top three finish to their campaign but dropped points with a 2-2 draw at home to Sekhukhune United and 1-1 stalemate away to Siwelele before succumbing to the Buccaneers.
The result left Zwane's team in fourth place with 39 points from 25 games, one point better off than Babina Noko in fifth and seven points behind Kaizer Chiefs in third place.
Reflecting on the result, Zwane admitted that the scoreline could have been worse had goalkeeper Olwethu Mzimela not made a number of key saves, including a second-half penalty intervention from Mofokeng.
"I think the goalkeeper kept us in the game. This is not a true reflection, the scoreline could have been worse," Zwane told SuperSport.
"Olwethu [Mzimela], did very, very well. But again, it boils down to one thing, if you don't put pressure on the ball, then you're giving a team like Orlando Pirates time and space for them to see the runners.
"We told them in our half, put them under pressure, force them to make a pass, then you will frustrate them. But if you allow Moremi to shape the ball inside the box to Mbuma. They had how many of those? You know, and we had a lot of good turnovers, even in first half like the one Athi [Maqokolo] missed early in the game.
"We had about six or seven of those moments where the players, they should go into goals. Just try. Have a shot at goals. See what's going to happen. They turn back.
"So yeah, it's one of those days where we were just nowhere near there. And coming here, it was always going to be difficult if we're not ready for it."
