Goals either side of the break from Mohamed Salah and Xherdan Shaqiri allowed the Reds to cruise through the second half and put the pressure on champions Manchester City ahead of this afternoon's derby at the Etihad Stadium.
Things may have gone differently had Fulham not seen a goal debatably disallowed moments before Liverpool's opener, but it was the same story of lapses of concentration in defence costing Slavisa Jokanovic's newly-promoted outfit.
The recalled Shaqiri, left out of Tuesday's Champions League defeat to Red Star Belgrade, went a little closer with his own effort from the edge of the box on 15 minutes, only a couple of yards wide after finding space in the inside right channel.
When the Reds did find the target, Fulham goalkeeper Sergio Rico proved tough to beat, standing up tall to block from Salah after some neat interplay with Roberto Firmino, and also denying the Egyptian's half-volley from a tight angle with a strong hand.
Fulham should have taken the lead on 23 minutes when Ryan Sessegnon latched onto a flick-on from Aleksandar Mitrovic to go through one-on-one with Alisson Becker, choosing to take on an early shot which he dragged away from the bottom corner.
Alisson was called into action moments later, and was fortunate that Joe Gomez was on hand to bail him out after he parried Andre Schurrle's powerful 25-yard drive straight back into the danger area, with both Mitrovic and Sessegnon sniffing for a rebound.
The Cottagers thought that they had taken a surprise lead when they worked a corner short for Tom Cairney to cross onto the head of Mitrovic, who glanced past Alisson and appeared to be at worst level with Andrew Robertson when the offside flag was raised.
Fulham switched off and within 20 seconds and two passes, Liverpool had raced up the other end to score as Alisson played the ball out to Alexander-Arnold, who released Salah to dart clear of Denis Odoi and slot through the legs of Rico on 41 minutes.
Jokanovic may feel doubly aggreived as the ball was still rolling when Alisson started the counter from a free kick, which referee Paul Tierney failed to spot, and the Londoners were out of the match when Liverpool doubled their lead eight minutes after the restart.
Robertson swung in a looping cross and Shaqiri was left unmarked at the back post to send a cushioned volley past Rico.
The on-loan Sevilla stopper kept the scoreline respectable with a good save from Roberton on 63 minutes, with the Scotsman a constant second-half threat on the overlap but unable to add a goal to his assist after being released by a well-timed Firmino through-ball.
Liverpool chose to stroll through the final 20 minutes as their unbeaten top-flight run at Anfield was extended to 27 games with a victory which will keep them top heading into the international break, if rivals Manchester United can do them a favour against City later today.
LIVERPOOL (4-2-3-1): Alisson; Alexander-Arnold, Gomez, Van Dijk, Robertson; Fabinho (Keita, 90'), Wijnaldum (Henderson, 69'); Shaqiri (Milner, 81'), Mane, Salah; Firmino
FULHAM (4-3-3): Rico; Christie, Odoi, Mawson, Le Marchand; Anguissa (Johansen, 84'), Chambers, Cairney (Seri, 63'); Sessegnon, Schurrle (Vietto, 78'), Mitrovic