Pirates opened the scoring in the 23rd minute through debutant Terrence Dzvukamanja. City then went down to ten men in the 33rd minute when Fagrie Lakay was given a straight red card.
And despite their numerical advantage and having several opportunities, the Buccanneers were unable to make the result more comfortable and could have been punished had Tashreeq Morris made contact with Abubaker Mobara's delivery in the closing stages.
"It was a difficult, hard game for us. Good structure from the opponent on defence and offense. They are a team that is always on fire on the counter-attack," Zinnbauer told SuperSport TV
"But it was the first game. The first game is not easy for any team and we saw the structure was good on defence but on offence not so good.
"We had a lot of chances. We could have decided the game before the 90 minutes. Usually we finish the game off. When you don't get the second goal, you get problems. Maybe you saw it yourself. We had a lot of counterattacks and chances but the final shot or final pass was not good. That's the next step we have to work on.
"Three weeks is not enough time to get the system in to the team. We have a lot of new players. The new players were not in the training sessions. Only one week together. And then you see the last step or moment is not hundred percent.
"It's a cup game. It's different to a league game. I'm sure next week when we play in the league we will have more structure."