Luis Suarez slotted in the home side's first in the 14th minute from Jose Enrique's pass and Daniel Sturridge doubled their advantage with a fierce strike three minutes later.
With seven minutes remaining before the interval, Steven Gerrard converted a penalty given for a Dean Moxey foul on Raheem Sterling.
Substitute Dwight Gayle headed in a consolation goal for second-bottom Palace in the 77th minute.
In third place, behind Arsenal, are Manchester City, who ended their difficult week on a high in yesterday's early kick-off as they came from behind to beat Everton 3-1 at the Etihad Stadium
City went into the contest off the back of last weekend's 3-2 top-flight defeat at Aston Villa and Wednesday's 3-1 Champions League home loss to Bayern Munich and they were behind in the 16th minute to a Romelu Lukaku goal.
But Alvaro Negredo equalised a minute later and Sergio Aguero put the hosts 2-1 up just before half-time, before the unlucky Tim Howard diverted City's third into the net in the 69th minute. The Everton goalkeeper tipped Aguero's penalty - awarded for Seamus Coleman's challenge on Pablo Zabaleta - on to a post but the ball hit Howard's head on the rebound and went in.
Fulham moved out of the relegation zone after posting their first win in six league games - and only their second of the season - with a 1-0 home victory against Stoke.
The three points were sealed by Darren Bent's left-footed strike with seven minutes to go.
Newcastle, who had suffered losses in their previous two league outings, also bounced back as they won 2-1 at Cardiff, with Loic Remy netting both of their goals to take his tally to five in his last three matches.
Remy drilled in his first on the half-hour mark having burst forward and then converted on the follow-up in the 38th minute when Scotland goalkeeper David Marshall parried Papiss Cisse's shot. The home side pulled a goal back 13 minutes after the break through Peter Odemwingie, but Newcastle were able to hold on.
In the day's other 3pm kick-off, Hull and Aston Villa played out a 0-0 draw at the KC Stadium.