Kenwyne Jones put the Potters into the lead on Wednesday night, but a Luis Suarez double in the second half ensured that the Reds move into the final eight.
Dalglish has praised the efforts put in by his team, especially after conceding the first goal.
“We were good as well as patient. We had six chances in the first half and we came in one down,” the Scot told Sky Sports.
“So great credit to the boys that they kept the belief, they kept at it and a fantastic goal for Luis followed on with a smashing header as well.
“I think [when] you come here you know you’re always going to have to defend and if you don’t match them you’re going to be in big trouble.
“And I think the boys matched them really well and they tried to play football when they got the opportunity and they did it.
“We’re very pleased with the result,” he concluded.
Liverpool needed the win after a disappointing 1-1 draw with Norwich last weekend, and now have a game against West Brom on Saturday to look forward to.
By Gareth McKnight