Sadio Mane scored the second on a counter attack breakaway following a stoppage time corner kick for Palace.
We look at five reasons why The Reds are genuine title contenders and have their best chance of a league trophy in 28 years.
Captain Luka Milivojevic has disputed referee Michael Oliver's decision to award Liverpool a penalty during Crystal Palace's 2-0 defeat at Selhurst Park. The strength, speed and determination of Roy Hodgson's team meant the visitors could not dare relax until the 93rd minute, when Sadio Mane raced away to seal three points. We don't know what will happen in the future (with transfer fees), but it's not important to us. "They are clear in their approach, playing a big number of long balls. I'm happy with that".
Liverpool legend Michale Owen hopes Egypt worldwide Mohamed Salah wouldn't move to La Liga side Real Madrid and eventually end his career at Anfield.
"Nobody thinks about it anymore and that's good because he's a player in this market, in the moment, he's at least worth it".
The score was 1-0 at the time, Milner having slotted in the penalty on the stroke of halftime after Salah tumbled under a challenge from Mamadou Sakho.
In the end, Liverpool takes care of business and picks up a well-deserved result, and now the Reds have six points and six goals in two matches. "We knew before the game it would be", Liverpool defender Virgil van Dijk said.
"I take some comfort and pride in the team's performance, but it won't change my disappointment and anger", he said. "We should have come in at 0-0 at half-time and the second half would have been even tougher".
No official explanation has yet been given by Palace or the Premier League but Ayew's exclusion from the team's squad for the opening weekend win over Fulham raised eyebrows. "Our target is to be a hard opponent".
Hodgson brought in Kouyate and Max Meyer to add further reinforcements in central midfield, yet on the evidence of the Fulham match these two new signings may have to wait in order to break into the starting lineup.
"If people can't see that then I'll be the pariah who sees only what he wants to see, and if all you guys see it as something different then I'll accept it".