An inspired performance from Swansea goalkeeper Michel Vorm, who made two stunning late saves to deny Luis Suarez and Glen Johnson, ensured Dalglish's side went into the international break on a low note.
Liverpool remain unbeaten at home, but have now dropped eight points in drawn games to leave them in sixth place.
Yet Dalglish is convinced Liverpool have enough quality to reverse their slide once they return from a two-week break.
"We just need to keep working on it and keep getting in there. But we haven't got in there as often as we have done in the past games. The problems there are problems we can solve ourselves," he said.
"We have to look at ourselves and expect better performances than what we produced there."
Andy Carroll was guilty of one especially bad miss but Dalglish refused to blame the misfiring former Newcastle forward.
"Anything that we were good at, we didn't do too much of, passing and moving, we didn't do too much of that," he said.
"The keeper was tested a few times and usually we are missing the target. That's 11 times we have hit the post or bar this season.
"If we had scored early on it might have changed the way the game went but we never scored so it is hypothetical.
"No-one has ever in the history of football deliberately missed a chance."
Liverpool lacked the drive of Steven Gerrard, but Dalglish said the midfielder had no further problems after checks on an ankle infection.
"He's fine. Everything been checked and the figures are hugely encouraging and well on his way to recovery," he added.
Where Gerrard will make a difference to Liverpool on his return, Dalglish has no doubt Swansea will avoid relegation with repeat performances like this, which brought applause from the home crowd at the end.
And Swans boss Brendan Rodgers agreed that his side are growing as they adapt to life in the top flight.
"To put in a performance like that at a big club with top players fills me with great pride," he said.
"I felt the first games we had away in this league was about finding out about this league and it's quality. We have now grown into it and we are starting to do well.
"When you come to Anfield and perform like that, it gives you psychologically a massive boost going forward.
"Good players are the best learners. If you learn and learn quickly you can succeed."
Rodgers praised Swansea for a collectively outstanding performance, but he has been especially delighted with Vorm, who has impressed since his arrival from Utrecht in August.
"When he has been called upon he has great agility," he added. "There are probably not too many goalkeepers come to Anfield and keep clean sheets.
"He made a terrific save at the end and I thought we were deserving of that as a team collectively and Michel was a part of that."
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Source: AFP