But despite City largely controlling the game, it took a late third goal by their substitute Ilkay Gundogan to ensure there could be no repeat of United's dramatic comeback to win 3-2 in the same fixture last season.
The half swung Liverpool's way in dramatic fashion four minutes before the break.
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.
Sadio Mane almost doubled the lead after 53 minutes with a shot from 25 yards parried behind by Rico, but Shaqiri was able to score after the resulting corner with a controlled finish from Andrew Robertson's cross.
"I'm not entirely sure his left foot is as good as 1970s Liverpool legend Alec Lindsay's but the cross he provided for Xherdan Shaqiri to score looked pretty special".
What does it mean?
Fulham's sense of injustice will only have been heightened when replays suggested Mitrovic's effort could well have been allowed to stand, with the London club seemingly on the wrong end of the kind of decision that so often goes against teams at the bottom of the table.
Given the freedom to roam across the front, Shaqiri was a consistent threat to a Fulham defence that has conceded 31 times in 12 league matches. His visit to Anfield was no exception: he missed two chances in the first half, one of which was a sitter, and offered next to nothing after Fulham fell behind. He's second runner from the top of the picture on the shoulder of the Liverpool defenders.
- Liverpool have kept 10 clean sheets in 14 Premier League games at Anfield in 2018; as many as they'd kept in the last calendar year (10 clean sheets in 21 home league games).
Chelsea's manager Maurizio Sarri set a record in the Premier League era by completing his first dozen games without a defeat, but he would have expected more than a goalless draw from the home game with Everton.
The win takes Liverpool's points tally to 30 after 12 Premier League games - their joint-best total at this stage of a Premier League season.