Mohamed Salah helped Liverpool back to the top of the Premier League with a goal that came 14 seconds after Fulham had an effort controversially disallowed for offside.
Replays showed the decision on Aleksandar Mitrovic's header was borderline with Liverpool left-back Andrew Robertson appearing to be in line with the Serbia forward.
But as the Fulham players celebrated, Liverpool goalkeeper Alisson initiated a lightning counterattack that Salah, who ran unopposed from halfway, fully capitalised on.
Xherdan Shaqiri's cushioned finish in the second half confirmed victory for the Reds who equalled their best start to a Premier League season which came in 2002-03.
Chelsea, who face Everton (14:15 GMT) and Manchester City, who play neighbours Manchester United (16:30 GMT), will go back above Liverpool if they both win later today.
Up until Salah's goal it had been tough going for the hosts who started well but were then frustrated by a well-marshalled Fulham side who remain bottom of the division.
Slavisa Jokanovic's team had arrived at Anfield with the worst defensive record in the division but with on-loan Sevilla goalkeeper Sergio Rico also impressing, they kept the hosts at bay for long periods and also created several promising chances.
Ryan Sessegnon and Mitrovic both spurned good opportunities to score while the teams were level.
But once ahead, Liverpool – who have an unenviable record against the bottom three Premier League clubs under Jurgen Klopp – rarely looked like relinquishing their advantage as they consigned the visitors to a sixth consecutive league defeat.