Palace dominated possession against the Cypriot cup holders, but were hit by a sucker punch not long after half-time when Riad Bajic smashed home a powerful first-time finish.
There were plenty of chances for the Eagles, with Jean-Philippe Mateta, Eddie Nketiah and Daniel Muñoz all going close.
But it was not to be, as Oliver Glasner's team drew a blank for only the fourth time this season.
"The easy answer is that we didn't score from the chances we had." Glasner said. "We made one mistake and they punished us. I think that's the summary.
"We were against a very deep block and it's not so easy to create 10 clear cut chances. But at the end we had 13 or 14 finishes inside the box and we were not clinical enough to take the lead, or score enough goals.
"They had in total I think four shots and we made one mistake. So, congratulations to Larnaca. They were very efficient.
"For us there's a little bit of deja vu. We have had games where we create many chances, but don't score enough and then get punished.
"That is what we have to learn and what we have to improve."