Match ends, Crystal Palace 0, Liverpool 2.
Liverpool manager Jurgen Klopp said he was "not interested in sending a statement" to his side's title rivals after victory over Crystal Palace gave them a second successive Premier League win.
The Reds - who finished 25 points behind champions Manchester City in fourth last season - were worthy winners at Selhurst Park and join five other teams, including City, on six points after two games.
James Milner gave them the lead with a penalty after Mohamed Salah was brought down by Mamadou Sakho in the closing seconds of the first half.
Palace had young defender Aaron Wan-Bissaka sent off with 15 minutes left after he brought down Salah when the Egyptian was through on goal.
And Sadio Mane wrapped up the points in injury time when he ran from his own half before rounding Wayne Hennessey to score.
The Eagles had chances, and Andros Townsend curled an effort on to the bar after a mistake by the otherwise impressive Naby Keita.
"It's very early," said Klopp, when asked whether the result confirmed Liverpool's pre-season status as Manchester City's closest title rivals. "I couldn't care less really. We are not in a race with other Premier League teams each weekend.
"It is too strong to say anything after two match days. You can discuss whatever you want - just do it without us."
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Hodgson 'angered' - but were big decisions right?
Palace manager Roy Hodgson was furious with the penalty decision - but accepted Wan-Bissaka's red card.
For the penalty, former Liverpool defender Sakho put his arms on Salah and then twice appeared to attempt to trip the forward, who went down.
"We probably did enough to get a result," Hodgson told BBC Sport. "I don't think it was a penalty. I'm angered that a good result was taken from us."
But BBC Radio 5 live's co-commentator at Selhurst Park, Chris Sutton, felt it was a penalty. "There was enough contact," he said. "I can understand Palace fans are frustrated but it's bad defending."
The Premier League is the only one of Europe's top five leagues not to use video assistant referees this season - but even if a VAR official or referee Michael Oliver had the option of watching it again, it may not have been overturned.
The same was true when Wan-Bissaka clipped Salah before he could reach the penalty area. The Egyptian had possibly lost control before the tackle came, and some fans were not convinced about the decision, but Sutton said the referee had "no choice".
Hodgson agreed. "Wan-Bissaka catches him," he said. "I don't think he tried to pull him down but there was contact and he pulled him down. It was the right decision. He was the last man and did his best to rescue the situation. In another game he might have got a toe on the ball. I don't blame him at all."
Liverpool find resolve as Klopp warns: 'Run or I'll kill you!'
Liverpool are being widely tipped to sustain a title challenge this season but they were never likely to follow up their 4-0 opening-day win over West Ham with a similar result at Selhurst Park against a team in good form.
But they dominated, having more possession than the Eagles, who were on the back foot for most of the game.
Salah, who scored 44 goals last season and once on the opening day, did not look at full sharpness - but he was still involved in all of Liverpool's best moments.
The Egyptian was fouled for the penalty and the red card, had a couple of shots blocked and missed with two chips over Hennessey.
And he also set up Mane's late goal after Palace committed players forward for a corner, Salah playing the ball to his Senegalese team-mate, who ran 50 yards before going around the keeper to tap into an empty net.
"In these moments the fuel is really low and maybe the players need a bit of help from an angry manager - 'run or I will kill you' - and they did that with a fantastic counter-attack," said Klopp, asked about his tense demeanour in the final few minutes.
Palace's unbeaten run comes to an end
The Eagles, who were unbeaten in seven Premier League games going back to a 2-1 defeat by Liverpool on 31 March, worked hard and had chances but were second best against talented opposition.
Their biggest chance came when Townsend got the ball outside the box and sent a brilliant effort on to the crossbar, with Alisson beaten.
Luka Milivojevic also went close with a free-kick, and Wilfried Zaha, who signed a new five-year deal this week, had a low shot saved in the second half.
Christian Benteke never looked like adding to his two goals in 2018, with one shot from 20 yards flying wildly off towards the corner flag.
Full-back Wan-Bissaka played well and was involved in a good duel with Mane on Palace's left-hand side - and he was applauded off the pitch by home fans after denying Salah a chance to score with his red-card tackle.
Hodgson brought on summer signing Max Meyer for a late debut, but the German was not able to have an impact.
"I thought we pushed them very hard," said Hodgson. "We should have come in at 0-0 at half-time and the second half would have been even tougher. We gave them enough problems and we can consider ourselves unlucky to lose 2-0.
"I'm more interested in the fact we played really well [than talking about the referee]."
Man of the match - Virgil van Dijk (Liverpool)
"I don't know a lot of centre-backs in the world who can deal with Christian Benteke like Van Dijk did," said Klopp after the game.
Stats - Mane scores against Eagles again
- Mane has scored six goals in his eight Premier League appearances against Palace, more than he has against any other side.
- Liverpool won their 34th Premier League game played on a Monday - only Man Utd (43) and Arsenal (36) have more victories.
- The Reds have lost only one of their past 10 Premier League games, winning six and drawing three.
- Palace have won just two of their previous 20 Premier League games played against 'big six' sides at Selhurst Park (D2 L16).
- Hodgson suffered his 100th defeat in the Premier League as a manager (W87 D65).
- Wan-Bissaka is the youngest Palace player to be shown a red card in the Premier League (20 years, 267 days).
- Milner extended his record for the most Premier League games scored in without losing any of them (48 games: W38 D10).
- He has scored 13 of the 15 penalties that he has taken in the Premier League, including nine of his 10 for Liverpool in the competition. Milner is the first player to score eight consecutive Premier League goals from penalties.
Liverpool host Brighton on Saturday (17:30 BST), a day before Palace are at Watford (13:30 BST).
- 29Wan-BissakaBooked at 75mins
- 3van AanholtBooked at 42mins
- 10TownsendSubstituted forWardat 79'minutes
- 15SchluppSubstituted forMeyerat 83'minutes
- 17BentekeSubstituted forSørlothat 70'minutes
- 14J Ayew
- 66Alexander-ArnoldBooked at 52mins
- 4van Dijk
- 7MilnerSubstituted forHendersonat 67'minutes
- 8KeitaSubstituted forLallanaat 87'minutes
- 9FirminoSubstituted forSturridgeat 90+4'minutes
- Michael Oliver
- Shots on Target
Second Half ends, Crystal Palace 0, Liverpool 2.
Corner, Crystal Palace. Conceded by Joseph Gomez.
Substitution, Liverpool. Daniel Sturridge replaces Roberto Firmino.
Goal! Crystal Palace 0, Liverpool 2. Sadio Mané (Liverpool) right footed shot from the right side of the box to the centre of the goal. Assisted by Mohamed Salah following a fast break.
Corner, Crystal Palace. Conceded by Virgil van Dijk.
Foul by Adam Lallana (Liverpool).
Joel Ward (Crystal Palace) wins a free kick on the right wing.
Hand ball by Wilfried Zaha (Crystal Palace).
Substitution, Liverpool. Adam Lallana replaces Naby Keita.
Attempt missed. Roberto Firmino (Liverpool) right footed shot from outside the box is high and wide to the right. Assisted by Mohamed Salah.
Attempt missed. Max Meyer (Crystal Palace) right footed shot from the centre of the box misses to the right. Assisted by Wilfried Zaha with a cross.
Corner, Liverpool. Conceded by Wayne Hennessey.
Attempt saved. Mohamed Salah (Liverpool) header from the centre of the box is saved in the top centre of the goal. Assisted by Sadio Mané.
Substitution, Crystal Palace. Max Meyer replaces Jeffrey Schlupp.
Attempt saved. Sadio Mané (Liverpool) right footed shot from the centre of the box is saved in the bottom right corner. Assisted by Mohamed Salah.
Substitution, Crystal Palace. Joel Ward replaces Andros Townsend.
Attempt saved. Naby Keita (Liverpool) right footed shot from outside the box is saved in the centre of the goal. Assisted by Andrew Robertson.
Corner, Liverpool. Conceded by James Tomkins.
Attempt blocked. Mohamed Salah (Liverpool) left footed shot from outside the box is blocked.