Raul Jimenez put Doherty through on 56 minutes and, despite the tight angle, the full-back was able to beat Wayne Hennessey at his near post.
Jordan Ayew failed to produce in the absence of the injured Christian Benteke and the Eagles now sit four points above the relegation place
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Although the home side began confidently in the first ever Premier League meeting between the clubs, Hodgson's team were wasteful in the final third.
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Palace created enough opportunities, and both Zaha and Aaron Wan-Bissaka provided trickery, but the team lacked a cutting edge once again as they slipped to 14th in the table.
