Cahill fell to the ground suddenly grasping his hamstring as Willian collected the ball on the right flank, before racing inside and passing for Olivier Giroud to convert the opener in the sixth minute.
Palace players protested as the former Chelsea defender limped off to be replaced inside the opening 10 minutes.
Lampard said “any team would continue in that situation”, while his Palace counterpart Roy Hodgson had no complaints, labeling it “very unfortunate”.
“I have sympathy, because I know Gary, we are friends, and you could see straightway that it was a bad injury the way he went down,” Lampard said.
“But then we had to play on and finish the action, no one is kicking the ball out in that situation.
“It is not one of those but you obviously feel sorry for Gary on an individual note. I know for head injuries the game would always stop but for injuries like that it doesn’t stop. That is pretty clear.
“Any team would continue in that situation. Willian is through on goal. If that wasn’t the situation the game would be stop-start all the time.
“That is why we give the referee the decision.”
Hodgson, who said he expects scans to show a tear in Cahill’s hamstring which would end his season, understood Chelsea’s stance on the matter.
“It would have been asking a lot I think (for Chelsea to stop),” the former England manager said.
“People react instinctively - you find yourself behind the defence in a really good position to create a goal chance, you carry on and take it.
“I think it would be unrealistic for me to suggest they should suddenly put their foot on the ball and stop and run straight to Gary and see how he is. Thats not the way football is.
“It certainly was a very unfortunate incident that happened so early in the game and in the end it cost us a point - because that was the goal they won by. They won by one goal, and that was it.”
The result keeps Chelsea in the box seat to finish in the top four and clinch qualification for next season’s UEFA Champions League, moving past Leicester City into third spot and five points clear of fifth-placed Manchester United who play on Thursday.
Jamie Vardy scored late to earn Leicester a 1-1 draw at Arsenal on Tuesday, but they’ll be looking over their shoulders as they bid to secure Champions League qualification.
Danny Welbeck volleyed a spectacular winner as Watford claimed a significant 2-1 win over bottom club Norwich City in their bid to avoid relegation.