The Reds booked their last-four spot at Wembley with a 2-1 win over Stoke at Anfield and minutes later were given a draw against the winners of Everton and Sunderland, who must replay at the Stadium of Light.
"We know we play Everton or Sunderland but whoever we get is going to be a difficult game," said the Scot.
"Merseyside will be hoping Everton get through and they can all have a day out again but it is a really difficult place to go and get a result at Sunderland.
"But that is not our problem - that is theirs to sort out."
Luis Suarez opened the scoring for Liverpool with an impressive long-range strike in the 23rd minute only for former Reds striker Peter Crouch to head in an equaliser three minutes later.
Stewart Downing's second goal of the season in the 57th minute clinched victory.
"It was as difficult as I expected. They started really competitively at a high tempo for first 20 to 30 minutes and we had to match them," said Dalglish of the performance.
"I think it was a fantastic performance in many ways for us: it was not the best in terms of how we passed the ball but the way we competed and put our bodies on the line set an example for everyone.
"It shows what it meant to them to get through this tie.
"We scored two really good goals from Luis and Stewart which just got us over the line."
Dalglish is confident the Football Association will make the right decision and ensure their semi-final does not clash with the 23rd anniversary of the Hillsborough disaster, where 96 fans died, on April 15.
Liverpool have always had special dispensation to avoid matches on that date and Press Association Sport understands the club have already contacted the FA to remind them of the significance of that date.
"We won't play on the 15th. The FA and UEFA have always been exemplary and very supportive, and that's great credit to them," said Dalglish.
"I don't think it's even worth a line in a paper that we might play on the 15th because we won't.
"We've never played on the 15th before, have we? So there you are. Why would they change now?"
Stoke boss Tony Pulis, whose side were FA Cup runners-up last year, was disappointed not to have given Liverpool more of a test.
"In the first half we started a bit sloppy and then got into the game but we are disappointed Suarez comes up with a bit of magic to give them the lead," he said.
"Luckily we got our goal and I thought we were lucky to not come in (at half-time) winning the game.
"Second half we gave them a leg up with the goal, which was a poor one to concede, but when you are playing against one of the top four or five teams in the country they punish you and that is what they did.
"But you can't fault their effort and honesty.
"Good luck to Liverpool, I hope they do well in the next round."