The Reds were forced to come from behind after Nikica Jelavic had capitalised on a Jamie Carragher mistake to give the Toffeemen a half-time advantage.
But an equally disastrous blunder from Sylvain Distin gifted Luis Suarez an equaliser before Andy Carroll silenced the doubters by ramming home the winner three minutes from time.
It means Liverpool will have the chance to add an FA Cup to the Carling Cup they won in February when they face either Chelsea or Tottenham at Wembley on May 5.
However, Dalglish is refusing to get carried away, saying: "It is called a work in progress.
"We have said that many times before and there is no need for us to change. After the Carling Cup I said we are not the finished article and we are saying the same thing now."
Carroll started strongly on Saturday, setting up decent chances for Jay Spearing and Martin Skrtel, but was ridiculed long and loud by the Everton fans a minute into the second half when he met a lovely Stewart Downing cross at the far post but powered a point-blank header wide.
Mortified by the miss, he buried his face in his shirt as he trudged back up field but to his immense credit he did not allow the setback to ruin his day and he kept plugging away.
Once Suarez had tucked in Liverpool's 62nd-minute equaliser Carroll, like all his team mates, found an extra spring in his step and he battled strongly with Everton's centre backs in a bid to force a way through their well-drilled defence.
In the 78th minute he dragged a low shot across goal and wide and soon after, looking a little desperate, tried his luck on the turn from almost 30 yards, again earning the derision of the blue end of the ground.
Carroll, though, had the last laugh three minutes from time when he leapt high to meet substitute Craig Bellamy's free kick and cleverly steered a back header beyond keeper Tim Howard.
"I've had some criticism but I've just kept on going," Carroll told ESPN. "To get the winner here is a great feeling.
"I believed in myself every day and got my chance here. I had a few chances and should have probably scored earlier but I kept at it and it was a great ball by Craig and I just had to score with that one and I did.
"It's the best feeling ever. We worked hard and getting the goal right there at the end was great," added the 23-year-old who also headed a stoppage-time winner in the 3-2 league victory over Blackburn Rovers on Tuesday.
Captain Steven Gerrard was happy to give his team mate praise.
"He doesn't hide and he takes criticism on the chin," Gerrard said. "That's what we bought him for, to score big goals, and he's delivered today."
Everton manager David Moyes offered words of sympathy to Distin after his error.
"Sylvain is really down," said Moyes. "He has been here a couple of times before and has won the FA Cup before with Portsmouth.
"He knows exactly what it means and how important it is. He has been great for us so far.
"But I said there would be little between the teams. I thought there would be defining moments and they were there for all to see."