Liverpool made it two wins on the trot as Dalglish celebrated his first Anfield victory since returning as manager.
The reds made one change from the weekend, with captain Steven Gerrard returning to the side from suspension. Lucas Leiva made way, after failing to recover from a thigh injury.
Liverpool started very brightly with a succession of chances in the first 15 minutes. Fernando Torres scored his 10th goal of the season before the male assistant referee ruled it out for offside. Replays suggested the decision was the wrong one.
Despite this error, the reds continued to press, a Meireles header was palmed wide, a Gerrard strike was saved and a Glen Johnson drive was tipped over the bar.
Eventually in the second half the reds grabbed a deserved opener, albeit from scrappier circumstances that the chances has suggested. Hangeland deflected a Torres shot on to the post and amidst the chaos Pantsil put the ball in his own net.
Fulham pressed late in to the game and the reds endured a nervy final 15 minutes which even saw Raul Meireles clear off the line.
However, on the balance of clear cut chances the reds deserved the three points in our view.
The reds move up to 7th, their highest position in the League since Rafa was Liverpool manager, and the goal difference is back at zero.
Liverpool must now focus on clawing the 5 point gap to 6th placed Sunderland.
Team: Reina, Kelly, Johnson, Agger, Skrtel, Poulsen (Aurelio 80), Rodriguez, Kuyt (Shelvey 66), Gerrard, Meireles, Torres
Bench: Gulacsi, Wilson, Cole, Pacheco, Ngog
Goals: own goal (52)
LFC Reds Man of the Match: tbc