The visitors started well and led through David Jones before second-half goals from Junior Stanislas, Danny Ings and Scott Arfield sealed their passage to the next round.
Dyche said: "It is an awkward one and I've seen some of the results already, these games can be awkward affairs but I think we made good work of it," he said.
"We started unbelievably for the first 20 minutes and I thought we could have
had possibly three goals.
"To be fair to York they came back and really put it on top of us for 20 minutes before half-time and then we managed to settle things in the second half and we ran away worthy winners.
"We had plenty of chances and I must give credit to their goalkeeper because
he made two really big stops.
"They didn't create that many chances, but they really put it on top of us. We
had to stand up for that period but over the course of the game our quality
shone and we got the right result."
York manager Nigel Worthington believes his side will learn some valuable lessons from the defeat.
"There are a lot of positives to take from that performance, I thought we
competed well but the thing that disappoints me the most is that we didn't start
in the first 20 minutes," Worthington said.
"There were no headers or tackles from us and they took advantage."