"I think there will be a Championship club there or thereabouts this year - and I hope it will be us," said the Eagles boss.
"A lot of good teams will get knocked out - there is a lot of pressure on the Premier League clubs this year, because they want to win it. Two or three more are taking it seriously.
"I wasn't bothered about the cup last year - which was a shame. But with not long to go in my career, I've told the lads I want a cup run.
"It was all about getting into the next round, but last week was so bad - at least there was some goalmouth action this time," added Warnock.
"I thought we were fired-up. I've told the lads that they'll soon forget the nights like this, when you make progress."
Foxes boss Nigel Pearson, an FA Cup finalist with Middlesbrough in 1994, also lost his centre-half Aleksander Tunchev to a knee injury.
He said: "The league is always our priority, but I'm disappointed nonetheless.
"I thought we gave a decent account of ourselves. The first half was even, but the first goal opened it up.
"But over the two games, I think we acquitted ourselves well."