Francesco Guidolin's side had the perfect start when Sigurdsson netted from a free kick, but in the 47th minute Dann scored to ensure that the two sides had a share of the spoils.
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The home side, who are fighting to avoid relegation from the Premier League, managed to put themselves in front when Sigurdsson sent the ball inside the right post after the team were awarded a free kick from distance.
For the rest of the first half, Swansea had the better chances as Andre Ayew came close when he aimed towards goal off a corner kick, but the ball fell narrowly wide of the right post.
Wayne Routledge also had a decent opportunity when he received the ball close to the penalty spot, but his shot just missed the target, inches from the left-hand post.
After the break, it took just two minutes for Palace to level proceedings when Damien Delaney flicked the ball on to Dann with his head before the defender fired the ball past the keeper.
The momentum swung in Palace's direction as Yohan Cabaye searched for a second with a curling effort on the edge of the box, but the ball brushed past the left post.
Swansea also had chances to nab the winner, one of which was produced from Modou Barrow, who darted away from his markers before letting one fly from 19 yards out, but the shot went over the crossbar.
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The best opportunity for the Swans came in the 84th minute when goalkeeper Wayne Hennessey denied Sigurdsson from scoring a second off another free kick.
The Icelandic midfielder's effort from just outside of the penalty area was on target, but he failed to get the ball through and that was the last real chance from either side as they were forced to settle for a draw.