Chelsea took the lead just six minutes into the game -- but there was a huge slice of fortune. Willian broke towards the box and was tracked by former Chelsea defender Gary Cahill, who suddenly slumped to the ground with an injury. That left Willian free to move into the area and pull the perfect ball back fro Giroud, who guided the ball home.
The visitors doubled their advantage just before the half-hour, Willian again the creator to set up Pulisic, who showed neat footwork before powering a shot past Vicente Guaita.
Chelsea looked to be cruising, and then out of nothing Wilfried Zaha powered Palace back into the game. Zaha picked the ball up 30 yards from goal and unleashed a curling drive that left Kepa Arrizabalaga clutching at fresh air.
Palace were having the better of the second half, but the sucker punch arrived in the 71st minute when two substitutes combined as Ruben Loftus-Cheek fed the unmarked Tammy Abraham to side-foot into the opposite corner.
One minute later, and Palace were back in it again. Patrick van Aanholt found space in the area and squared the ball for Christian Benteke to tap into an empty net.
Mason Mount went close to making the game safe, but once again Chelsea's suspect defending was their Achilles heel -- and Palace thought they had equalised in the fifth minute of added time when Scott Dann's header came back off the post and was cleared to safety.