"This time last year we were in a similar position, then went on a run to take us to a play-off position by January. I feel strongly that this is very possible yet again," said the departing former Scotland coach.
"I know the team is in the hands of an extremely capable coach and I am sure that Billy is the man to bring success to the club."
Preston chairman Derek Shaw said of Brown's departure: "I would like to put on record my sincere thanks for all Craig's efforts and it is sad for myself and everyone connected to the club that such a fine ambassador will be leaving.
"This is a very difficult decision as these people become very close, but we have agreed, in the best interests of Preston North End, that it is time for a change."