Ben Chilwell also got his first for the Blues while Zouma also got on the score sheet.
It was a game of few chances in first half, with perhaps the best opening falling to Tammy Abraham, who could not take a pass from Jorginho in stride in front of goal.
Palace were resolute at the back, but not much of an attacking threat as it ended goalless at first half break.
A mistake at the back from Sakho allowed Chelsea to take the lead.
Hudson-Odoi tried to beat the challenge of the Frenchman, who stopped his progress. However, he slices his clearance straight to Azpilicueta.
The Spaniard lifts a cross into the box where Abraham diverts it into the path of Chilwell. The left-back hammers a left-footed strike past Guaita into the net to notch his first goal for the club.
66 minutes on, Zouma doubled the lead for the hosts. Chilwell sees his corner repelled, but the Blues recycle possession. The England full-back gets a second chance and puts his delivery on the money.
Zouma gets above Kouyate and glances his header away from Guaita into the top-right corner.
Jorginho made it 3-0 for Chelsea 78 minutes on the clock. The Italy international held his composure to score from the spot kick.
He does his trademark hop on his run-up before placing a right-footed strike into the right side of the net away from the dive of Guaita.
4 minutes later, Jorginho scores again to complete the rput for the Blues.
He notched his second with another cool penalty. It’s a carbon-copy of his first spot-kick, placing his effort to the right away from Guaita.
Frank Lampard will be the happy Manager over Roy Hodgson with his team’s performance.