Dann came on as a half-time substitute at Stamford Bridge following an injury to fellow centre-back James Tomkins.
But the 30-year-old had to be withdrawn 15 minutes later after twisting his knee while blocking a Diego Costa shot.
Palace boss Sam Allardyce said the injury was "more serious than we first thought".
"He'll be in a brace for two or three weeks and then we'll go from there," he added.
Allardyce said the injury suffered by Tomkins was "not so bad", but he would not be available for his side's next match.
Palace, who are 16th in the Premier League, four points clear of the relegation zone, travel to Southampton on Wednesday (19:45 BST).
Crystal Palace captain Scott Dann will be out for at least "a few weeks" after suffering a knee ligament injury in Saturday's 2-1 victory at Chelsea.