Pirates were off their game in the first half as Zinnbauer noted but in the second half the Bucs found their feet and produced a number of chances which they failed to capitalise on.
"The build-up was not enough," Zinnbauer told SuperSport TV.
"The high press from Swallows was good and we didn't find a solution. This was the biggest problem at the beginning. The mistakes as well. This is a problem. Individual mistakes are not easy for the whole team.
"We got better in the game. Second half was much better. We had opportunities to score in the first half as well. You need a little bit of luck.
"Yeah, a lot of opportunities in the second half. But this is always the problem. The playing style we wanted was there in the final third. But you need the shot at goal. We needed more confidence to go for goal more.
The Pirates head coach also defended Bogani Sam, who was deployed in the left-back position.
"We will work on it. It was a bad start in terms of result. Performance got better. You could also see [Bongani] Sam was the first time in his position. It wasn't only Sam. The build-up in the midfield was not good enough in the opening half," he continued.
"We changed players to help with the build-up. In the midfield you need more movement. We want the lead, we want to dominate. I don't know how many chances we had to score. If you don't score, you cannot win a game.
"We have a week now. We will get other players back. We have other players as well. Going forward, it's a bad result. Next week is another week and it's coming."