Liverpool striker Carroll, who has struggled since a big move from Newcastle United a year ago, did not find the net but used his physical prowess in the area as Daniel Agger headed in the first and then set up Dirk Kuyt for the late winner.
A contender last year for England's Euro 2012 squad, the 23-year-old Carroll has dropped well down the pecking order and has been in and out of the Liverpool side despite fellow forward Luis Suarez serving a long ban.
"Everybody knows Andy's prowess in the air, we've tried our best to develop that further and we'll also work and be patient with him," Dalglish told ITV Sport.
"His commitment was superb today so as long as you've got the right attitude and the right commitment we've got a chance of working with it."
The always nervous 21-year-old was crowded out in the box for Agger's header from a corner and could have made a better attempt as Kuyt thrashed the ball into his near post.
"We didn't do well. [There were] too many defenders in and around the goalkeeper for the first one and the second one is just one of those lucky breaks you get," he said.
"I'm pleased with my players, they performed very well and dominated the game."
