The Croatian played 90 minutes at the KCOM Stadium as the Blues secured an ultimately hard-fought 2-1 victory to book a place in the hat for tonight's fifth round draw.
"It was a tough game, as we knew that it would be, but we need to score in the first half two more goals," the 25-year-old said afterwards.
"That's our problem in this moment but obviously we're happy to go through.
"It was because we didn't score on time - we had some chances and we didn't use them. In training, we are working on it but we are still a young team, a new team, and we need to improve a lot. We need to take our chances, which in this moment we are not doing great, but we have time to improve and I am sure we will do."
For Hull's goal, Kovacic was criticised for breaking the structure of the wall with his jump and he was willing to accept fault.
"They scored a goal from a free-kick, which was my mistake," he admitted to chelseafc.com. "I did a little bit of a strange jump and the ball touched me. Sometimes that can happen but the most important thing is we got the win and we are through.
"The FA Cup is a great competition and we want to win it."