Lewis McGugan slotted home the spot-kick after Kevin Long collided with Henri Lansbury at the end of the game at Turf Moor, but Dyche felt the full-time whistle should already have blown.
"I feel sorry for our fans, because yet again they've witnessed an incredible situation where we can't just win against an opposition, we've got to beat everything else," said Dyche, whose side had taken the lead through Junior Stanislas' second-half goal.
"I'm disappointed in the whole referee's performance today and I'm sick of talking about it because I shouldn't have to.
"But it's not the first time as we know; it's been a constant issue since I've been at this football club. I don't know what it is, but that isn't good enough today.
"I had a word with him [Miller] and he said he was more interested in the fact we were late for kick-off. I was really amazed he when he told me that.
"The timing issue at the end inevitably affects the game. It probably is a penalty but that's not the point. The game's dead, it's over, we've won.
"The lads have given everything and like I say it's very difficult when you've got to beat more than just the opposition."
Dyche believes that the Premier League should help to improve the standard of refereeing in the lower leagues.
"I've said it many times and I'll say it again; there needs to be more support for these referees, the only people who can do it are the Premier League because they're so wealthy," added Dyche. "The Premier League must start funding the development of referees."