Goals in each half from Mohamed Salah and Xherdan Shaqiri secured all three points on Sunday and moved Jurgen Klopp's side back to the top of the Premier League table, at least temporarily.
Liverpool were held by Arsenal in their last league outing before a 2-0 loss to Red Star Belgrade this week, and they looked some way short of their best in a first half in which Fulham had the best chances.
However, the game turned within the space of 14 first-half seconds, when Salah struck after Aleksandar Mitrovic had a header ruled out for offside, and Shaqiri's fine finish made the win secure against a Fulham side who have now lost seven games in a row and remain bottom.
Liverpool's early dominance of the ball led to a personal duel between Salah and Sergio Rico, the Fulham goalkeeper twice making good saves to deny forward.
Shaqiri twice messed up shots from good positions and the home side's profligacy should really have been punished, with Ryan Sessegnon dragging wide when clean through on goal before Alisson almost parried an Andre Schurrle shot into his path.
The half swung Liverpool's way in dramatic fashion four minutes before the break. Mitrovic headed in Tom Cairney's cross only to be contentiously ruled offside, Alisson took a quick free-kick and Trent Alexander-Arnold sent Salah clean through to slot past Rico.
It was a heavy blow to Fulham and the game was effectively beyond them eight minutes after the restart, when the unmarked Shaqiri expertly volleyed home a pinpoint cross from Andy Robertson.
Robertson was denied a third by a good Rico save, but Liverpool had few problems seeing out a ninth league win of the season, while concerns for Slavisa Jokanovic look to be building.
What does it mean? Reds ride their luck back to the top
This was not a hugely improved performance from Liverpool after their poor display against Red Star Belgrade, but it was enough to see off a Fulham side bereft of confidence.
Salah and Shaqiri showed a clinical touch after a couple of first-half scares, and Klopp will doubtless be pleased to go back to the top of the table at least before Manchester City and Chelsea each play later.
The visitors should have taken the lead through Sessegnon early on and will point to Mitrovic's disallowed goal as a turning point, but the way they failed to respond to going behind will be a major concern for Jokanovic. They remain bottom, two points behind Huddersfield Town.
Shaqiri a menace in free role
Given the freedom to roam across the front, Shaqiri was a consistent threat to a Fulham defence that has conceded 31 times in 12 league matches. He could have scored from distance in the first half and fully deserved his goal when it finally came.
Sessegnon falling short
He is still only 18 and looks set for a fine career at the highest level, but Sessegnon has found the Premier League a challenge. His visit to Anfield was no exception: he missed two chances in the first half, one of which was a sitter, and offered next to nothing after Fulham fell behind.
Key Opta Facts
- Liverpool have picked up 30 points from their opening 12 Premier League games (W9 D3 L0) – their joint-best total at this stage in the competition (level with 2002-03).
- Fulham have conceded 31 goals from their first 12 Premier League games this season; only Barnsley in 1997-98 conceded more goals at this stage of a Premier League campaign (35).
- Liverpool have kept 10 clean sheets in 14 Premier League games at Anfield in 2018; as many as they’d kept in the last calendar year (10 clean sheets in 21 home league games).
- Mohamed Salah has scored against 19 of the 21 Premier League opponents he's faced with Liverpool, failing only against Man Utd and Swansea.