On Sunday evening, Liverpool were without the injured Mohamed Salah, but still had Sadio Mane and Roberto Firmino available and they were decisive as Crystal Palace went close to costing Jurgen Klopp's side two points, reports Xinhua news agency.
The home side had a goal ruled out in the first half before Mane put Liverpool ahead two minutes after the break.
Wilfred Zaha looked to have saved a point for Palace with just 8 minutes remaining, but Firmino was on hand five minutes from time to maintain Liverpool's triumphant start to the season.
Manchester City suffered a painful defeat in Anfield two week ago which saw Pep Guardiola's side drop to fourth in the table, but they bounced back with a thrilling win at home to Chelsea.
The game was fast and open and perhaps showed up the defensive flaws which could ultimately cost both sides this season. N'Golo Kante put Chelsea ahead in the 21st minute, but City reacted through Kevin de Bruyne and Riyah Mahrez whose 37th minute goal was the last of the game, although Sergio Aguero later rattled the upright for City.
The other big story of the day was Jose Mourinho's return to football and the Portuguese coach was back with three points as Tottenham secured their first away win with a 3-2 win over West Ham.
The final score makes the game seem closer than it actually was, as Tottenham led 3-0 by the 49th minute as the impressive Heung-Min Song, Lucas Moura and Harry Kane made the points safe against a rival that has now taken just two points from a possible 21 to leave coach Manuel Pellegrini looking over his shoulder.
Arsenal's Unai Emery must feel the walls closing in around him as his side were held to a 2-2 draw at home to Southampton.
Sunday ended with perhaps the most entertaining match of the weekend as Sheffield United and Manchester United shared six goals at Bramall Lane. John Flack and Lys Mousett put Sheffield 2-0 up with goals after 19 and 52 minutes only for Manchester United to bounce back with three goals in seven minutes as Brandon Williams, Mason Greenwood and Marcos Rashford put them ahead with goals between the 72nd and 79th minute.
The points looked as if they would go back to Manchester but Ollie McBurnie volleyed past David de Gea in the last minute to assure the a share of the points.
( With inputs from IANS )