Daniel Sturridge's fourth-minute goal was enough to secure all three points for the Reds, who have yet to concede a goal in the league, and preserve their 100 per cent start.
The former Chelsea man had already given warning of his threat when he seized on the loose ball after Jordan Henderson had harassed Ashley Young into a mistake. Rio Ferdinand turned away his shot on that occasion.
But when Daniel Agger leapt highest at the corner, Sturridge was afforded far too much room inside the six-yard area to turn the ball home.
Olivier Giroud scored the only goal as Arsenal beat Tottenham 1-0 in another of English football's great grudge matches at the Emirates.
With the Gunners enduring a frustrating time in the transfer market and Spurs joining the Premier League big spenders, many have predicted a changing of the guard in north London.
But that was not on show on Sunday as a swift counter from Aaron Ramsey and Tomas Rosicky saw the ball played out to Theo Walcott down the right.
The England forward cut a pass towards the six-yard box - where Giroud arrived ahead of Spurs captain Michael Dawson at the near post to convert a lovely left-footed finish.
Tottenham's bad afternoon was further compounded as summer arrival Etienne Capoue was carried off midway through the second half.
In the day's other game Swansea claimed their first points of the season as they beat West Brom 2-0.
Having lost to Manchester United and Tottenham, Michael Laudrup's side were in need of a result to lift them off bottom spot and duly delivered with Pablo Hernandez teeing up Ben Davies for the opener before applying the finishing touch in the 83rd minute.
On Saturday, Manchester City bounced back from their shock defeat at Cardiff with a routine 2-0 victory over Hull in the Barclays Premier League at the Etihad Stadium.
On the back of a clean sheet from the heavily-criticised Joe Hart in the wake of his poor performance against Cardiff, second-half goals from Alvaro Negredo and Yaya Toure were enough to see off the Tigers.
For Negredo, a £20million summer buy from Seville, it was his second goal in two matches after his late strike in south Wales, and again after coming off the bench.
The Spaniard opened the scoring in the 65th minute with a close-range header, but it was not until the 90th City finally made the game safe via a stunning free-kick from Toure.
Remarkably, as with the opening 45 minutes at the Etihad, the four 3pm kick-offs also failed to produce a goal.
Looking to follow up on their triumph over Manuel Pellegrini's side, Cardiff barely mustered an opportunity on home soil against Everton.
Instead, it was the visitors who were denied a shot at taking the lead when Leighton Baines was brought down in the area by Gary Medel, only for referee Anthony Taylor to wave play on.
There should arguably have been a penalty, too, at Carrow Road when Bradley Johnson handled a shot from Southampton's Adam Lallana, but Howard Webb was unmoved despite a clear sight of the incident.
To even matters up, Webb later dismissed shouts from the Canaries when a Ricky van Wolfswinkel header hit the hand of Calum Chambers.
Lallana later tried his luck with another cracking drive, only to be denied by the post.
At Upton Park, West Ham goalkeeper Jussi Jaaskelainen was one of the busiest players on the pitch in an opening half dominated by visiting Stoke.
Jaaskelainen was forced into making key saves from Charlie Adam, from a 25-yard free-kick, Marc Wilson from a long-range volley and then Jonathan Walters on the follow up.
In the other game at St James' Park, Newcastle had the edge over visiting Fulham, but despite a couple of chances for Papiss Cisse, there was again no breakthrough.
Saturday's Premier League Results
Crystal Palace 3-1 Sunderland
Cardiff 0-0 Everton
Newcastle 1-0 Fulham
Norwich 1-0 Southampton
West Ham 0-1 Stoke
Man City 2-0 Hull
Sunday's Premier League Results
Liverpool 1-0 Manchester United
Arsenal 1-0 Tottenham
West Brom 0-2 Swansea