It took a scrappy 52nd minute own goal from John Pantsil, who sliced the ball into his own net after Fernando Torres shot deflected off the post, to secure the three points and Liverpool's second consecutive Premier League win after the weekend victory over lowly Wolverhampton Wanderers.
The win lifted Liverpool to seventh in the standings with 32 points.
"We will take anything as any club will do," manager Dalglish told Sky Sports. "At times we played good football and deserved the bounce of the ball.
"I have seen three games and that is one of the few games we have had a little bit of good fortune."
Liverpool could have taken an early lead when a rejuvenated Torres latched onto a pass from Raul Meireles and poked it into the net only to be pulled back by the linesman's flag.
While Fulham keeper David Stockdale made fine saves to deny Meireles from eight yards and a long-range effort from Glen Johnson in a first half largely dominated by the Reds.
But after taking the lead, Dalglish's side failed to press home their advantage and allowed the visitors to maintain a foothold in the game.
Fulham pressed hard in the closing minutes and should have levelled when Moussa Dembele saw a shot from eight yards scrambled away by Reds keeper Pepe Reina.
"I think Fulham might feel a bit of injustice and feel they should have got something from the game," Dalglish added.
"I think in the first half we had six good chances and one chalked off that was really onside.
"But the lads put in a great effort and threw their bodies on the line because they didn't want to lose a goal and they have worked hard all season."
