The result means the Reds go into the international break in second place in the Premier League, just two points off top spot.
Here is what the media made of Sunday's match…
James Pearce, Liverpool Echo
The Reds went into the international break on a high after an ultimately routine victory over lowly Fulham. It was far from a ruthless demolition job but Jürgen Klopp's side overcame a sticky start to get the job done and put their Belgrade nightmare firmly behind them. Salah maintained his resurgence with his eighth goal of the campaign and Xherdan Shaqiri wrapped up the points early in the second half. This was only Shaqiri's fifth start but he's already made the leap from decent squad player to shining light. In his last six appearances for Liverpool he has contributed two goals and three assists. His claims for greater involvement simply can't be ignored, no matter who the Reds are facing. Shaqiri is a rare breed – strong as an ox but with the finesse of a ballerina. Crucially, he provides the Reds with the kind of creative spark they simply don't possess when he's not out there. Fabinho also caught the eye in the centre of the park, while Andy Robertson was outstanding as he rampaged down the left flank. Robertson hasn't grabbed too many headlines so far this season but he has quietly gone about his business and been a model of consistency. Living up to what he achieved during a dream debut campaign at Anfield was never going to be easy but he just keeps on delivering.
Neil Jones, Goal.com
It may have spluttered in midweek, but the Liverpool machine is back in business. Klopp called for "a reaction" from his players after Tuesday's Champions League shocker and he got the right one in the end. Bad as things were in Belgrade, you can't keep this team down for too long. In the end it was comfortable, the kind of afternoon for which the phrase 'routine home win' was invented. Fulham, the Premier League's bottom side, were overpowered in the Anfield sunshine. The 2-0 scoreline could have been more emphatic as the second-half wore on. This, then, is officially Liverpool's best start to a Premier League campaign. Never have they had more points after 12 games, never have they had a better goal difference, kept more clean sheets or conceded fewer goals. The team's progress, in all aspects, is there to see – whether they are playing 'City football' or not.
David Lynch, Evening Standard
When given a chance to start games for Liverpool, Shaqiri always produces. The Swiss added to a long list of key contributions in grabbing the Reds' second here, courtesy of a sensational volleyed finish at the back post. In the aftermath of Trent Alexander-Arnold's half-time withdrawal in Belgrade, it would have been easy for Klopp to leave Joe Gomez at right-back here and restore Dejan Lovren to central defence. But the 51-year-old clearly acknowledged that it is in his best interests to play his Scouse full-back into form as soon as he possibly can. Alexander-Arnold's crossing ability and comfort in possession are a direct result of his beginnings as a midfielder, and they so often prove vital to how the Reds play. The teenager wasn't perfect in this game, but showed that he is capable of putting poor performances behind him quickly. That skill will be vital to him delivering on his considerable promise.
Andy Hunter, Guardian
An eighth goal of the season from Salah plus a fine finish from Shaqiri took Liverpool back to the summit. It was the reaction their manager craved following the lame Champions League defeat at Red Star Belgrade in midweek, a fifth home league win of the campaign and the clean sheet ensured Liverpool's meanest start to a top flight campaign – five goals conceded in 12 matches – since 1979. There was no disputing the merits of Liverpool's second goal shortly after the restart. Rico saved superbly from Mané's rising drive. Alexander‑Arnold's subsequent corner was over-hit but Virgil van Dijk and Fabinho retained possession on the far side and fed the excellent Robertson. The left-back sent a first-time cross deep into the Fulham area where Shaqiri had ghosted away from Cairney and, with an expertly controlled volley, steered the ball beyond Rico. There was no way back for Fulham.
This story has been reproduced from the media. It does not necessarily represent the position of Liverpool Football Club.