Nicklas Bendtner's second-half strike was enough to condemn the Merseysiders to a third successive Barclays Premier League defeat which left them 10 points adrift of fourth-placed Arsenal.
However, Dalglish was refusing to be drawn into an assessment of their chances of closing that gap during the remaining 11 games of the campaign ahead of Tuesday night's derby clash with Everton.
Dalglish said: "We will say what we said at the very first game of the season - we will play all the matches, count the points up and see where we finish then.
"If we could maybe get fourth spot, it would be fantastic for the football club, but winning the Carling Cup was fantastic, getting into the sixth round of the FA Cup is more than where we have been for a little while.
"I don't think the performances that we have had, especially at home, have merited the number of points we have got.
"But we just need to keep going and do what we do well, keep our mouths shut, not feel sorry for ourselves and try to get as many points on the board as we can.
"It's a great credit to the lads who have come in, the seven new lads who have come in this season, to have put something on the board in their first season.
"There are not many places where there are that many changes and they get a success like that."
A distinctly ordinary contest was won 11 minutes after the break in bizarre circumstances as Pepe Reina suffered another nightmare not of his own making on Wearside.
The Spaniard, who was beaten by Darren Bent's beachball goal at the Stadium of Light in October 2009, looked to have been saved by the woodwork when Fraizer Campbell's shot hit the foot of the post after beating his dive.
But to his horror, the ball rebounded on to his back and hit the upright once again to allow Bendtner to fire home from close range.
In a game of few genuine chances, Bendtner's intervention proved decisive as the Black Cats saw out time in relative comfort with even the introductions of Steven Gerrard and Andy Carroll failing to pay dividends for Dalglish.
The Scot said: "I don't think it was much of a game and it was always going to be something, a bit of good fortune - I didn't think anybody was going to produce a bit of brilliance to win the game.
"We matched all the attributes that Sunderland have, and they have been on a great run, and I didn't think there was much in the game other than the bit of luck that decided the game."
Opposite number Martin O'Neill, who went into the game without suspended duo Lee Cattermole and Stephane Sessegnon and the injured Kieran Richardson, was delighted with a result which ended a three-game run without a league win and left them eighth, just two points behind Dalglish's men.
He said: "In a game of very few chances, it was great to get the goal and it eventually proved the winning goal in the match.
"With the changes that we had to make with influential players out of the team, I couldn't really have asked for much more. It was a great effort by the team."
However, victory came at a cost ahead of the FA Cup quarter-final trip to Everton with Bendtner leaving the field on a stretcher after suffering a knee injury which will be scanned on Monday.
O'Neill, who revealed there have been no talks over a permanent move for the on-loan Arsenal frontman, said: "We are not certain at all. He is obviously a concern for next week."