Two became three seconds before half-time, Mohamed Salah tapping in after a fine Shaqiri free kick had rattled the bar, and despite losjng ex-Southampton defender Virgil van Dijk to a knock, Joe Gomez replacing him, Liverpool were rarely threatened.
In the 10th minute Shaqiri cut in from the left to race onto Sadio Mane's perfectly-weighted throughball, hitting a right-footed shot off Shane Long onto Hoedt with scrambling goalkeeper Alex McCarthy unable to keep it out.
It may be too early to talk about title races, but Liverpool's form is ominous.
And the Reds' historic seventh win was marked in style!
Silva was on target within minutes, looping a header in, and City wrapped it up with a superb goal, Gundogan and Sterling linking and the former bending a brilliant finish into the corner.
Liverpool have won their opening five league matches and are second in the table, level on points with Chelsea, and Klopp says Salah remains a "threat" to the opposition. Fans will hope his withdrawal at half-time was not due to injury.
Liverpool take on Southampton in just under an hour and team news has been released.
Southampton will be without Danny Ings as he's not allowed to play against his parent club while Manolo Gabbiadini will face a late fitness test on a hamstring injury. That is more my mistake than his.
For Southampton it was a third loss in six league matches.
It denied the 26-year-old a much-needed brace, with Jurgen Klopp explaining after the game that he thought his star man was unlucky in front of goal.
Opta point out that Liverpool are unbeaten in their last 23 Premier League games at Anfield (W15 D8), their best run in the competition since August 2009, and have kept a clean sheet in each of their last seven Premier League home games.
- Matip scored his first Premier League goal at Anfield, in what was his 32nd such appearance.