Luis Suárez opened the scoring in the 24th minute, combining with Maxi Rodríguez to unlock the Trotters’ defence before curling his shot past Thomas Sorensen.
Former Liverpool striker Peter Crouch equalised for the visitors two minutes later, but Dalglish’s side moved into the last four courtesy of Stewart Downing’s venomous strike after 57 minutes.
“It was a fantastic performance from us in many ways,” said Dalglish after the game.
“It wasn’t the best we’ve ever passed the ball but the way we competed and the way we put bodies on the line was a great example.
“It showed people what it meant to them to try and get through this. It was difficult. Stoke started really competitively and at a high tempo.
“We had to match them, and although Pepe has not been too busy, they’re always a threat with throw-ins and set plays. There were two really good goals.”
Liverpool will face the winner of the quarter-final replay between Everton and Sunderland at the Stadium of Light on 27 March.
And although Dalglish admits an all-Merseyside semi-final at Wembley would be an exciting prospect, the Reds boss expects a difficult game whoever his side faces for a place in the final.
“We know we’ve got Everton or Sunderland, but whoever we get it’s going to be a difficult game,” he added.
“Merseyside will be hoping Everton get through and they can all have a day out again, but Sunderland is a really difficult place to go and get a result.”
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