But their shortcomings were exposed again when the unmarked Shaqiri wandered into the box to casually volley home Andy Robertson's left-wing cross.
Jurgen Klopp made some changes to his side following the midweek Champions League defeat to Red Star Belgrade with Shaqiri recalled in place of Adam Lallana to add some creativity in the middle of the park after a poor performance in Serbia.
Replays showed that the decision was tight but the ball was rolling when Alisson took the free-kick.
"I don't want to say Liverpool didn't deserve to win the game, they created chances and they can score the goal".
Liverpool FC will travel to Watford after the global break.
Liverpool broke the deadlock in controversial circumstances in their 2-0 victory over Fulham.
What does it mean?
Fulham should have taken the lead on 23 minutes when Ryan Sessegnon latched onto a flick-on from Aleksander 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.
Scott Parker is in contention to replace him should he be sacked if their struggles continue, but against that backdrop and the prospect of facing Liverpool's Sadio Mane, Roberto Firmino and Mohamed Salah, the 50-year-old Jokanovic has no intention of heading to Anfield to defend. They remain bottom, two points behind Huddersfield Town.
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.
- 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).
In six league games at home, Salah has scored in five of them - the only blank coming against title rival Manchester City.
United missed Paul Pogba through injury and offered little in attack in the first half, with Marcus Rashford a frustrated figure through the middle.
- Mohamed Salah has scored against 19 of the 21 Premier League opponents he's faced with Liverpool, failing only against Man Utd and Swansea.