Liverpool, however, might suggest they were due some luck after an incorrect flag had denied them a possible victory over Arsenal a week earlier.
After a minute's silence to mark the centenary of the Armistice in World War I, Liverpool started stronger. Alexander-Arnold's low cross eventually found Sadio Mane and he should have done better with a shot which drifted well wide.
Roberto Firmino, operating in the hole behind Salah, found space to feed Shaqiri, whose shot flew just wide of the post.
Liverpool manager Klopp had asked his team to rediscover the joy in their football following their shock midweek Champions League defeat by Red Star Belgrade, and it almost arrived 15 minutes in.
Salah and Firmino twice exchanged give-and-goes to cut Fulham apart but Sergio Rico was swiftly off his line to deny the Egyptian.
Liverpool continued to press, with Fabinho's fine diagonal headed over by Alexander-Arnold, while Salah's powerful drive from a narrow angle was turned away by Rico.
A rare mistake from Joe Gomez presented Fulham with the game's best chance midway through the half. The England defender failed to deal with Mitrovic's header, allowing Ryan Sessegnon to close in on goal but he somehow pulled his shot wide with only Alisson to beat.
Fulham were encouraged and a few minutes later Andre Schurrle's drive was spilled by Alisson, with a timely intervention by Virgil van Dijk required to clear the danger.
But Salah's opener allowed Liverpool to gain control and they underlined their dominance eight minutes into the second half when a short corner routine saw Andy Robertson's cross find an unmarked Shaqiri, who scored with a volley.