Fresh from hammering West Ham United in their opening encounter, Liverpool started with the same eleven that were so effective a week ago.
The Merseyside outfit thought they had gone ahead early on through Sadio Mane, but the effort was ruled out for offside after a foul on Wayne Hennessey.
Mohamed Salah was sent one-on-one with Hennessey with a perfect through ball, but the Egyptian shot well over the top on 23 minutes.
Andros Townsend went close at the other end, smashing a long range shot against the crossbar two minutes later.
Hennessey was called to action moments later to deny Naby Keita at the near post after a nicely-worked move by Liverpool.
A good piece of defensive work from Salah to hold off his marker allowed him the opportunity to cross into the box for Andrew Robertson, who unfortunately fired a first-time strike over the top.
Liverpool were handed a penalty just before halftime after a foul on Salah inside the box. James Milner made no mistake from 12-yards to give the visitors a lead at the break.
A brilliant free-kick from Luka Milijojevic drew Alisson to palm the attempt around the post following 53 minutes of action.
Wilfred Zaha brought down a cross inside the box to fire towards Alisson, but the end outcome failed to cause any problems to the Brazilian who saved easily on 73 minutes.
Palace were reduced to ten men after Aaron Wan-Bissaka brought down Salah on the edge of the box 15 minutes from time.
In the closing stages, Hennessey made his first real save of the second period, tipping Salah’s header over the top on 84 minutes.
There was absolutely nothing he could do to stop Mane from adding a second on the counter attack at the end of added time to round off back-to-back triumphs for the Reds.