They trail Tottenham by five points with only nine games left in the season, and may have to settle for a Europa League place come the end of the season.
The pressure will be on both Antonio Conte and Roy Hodgson as they search for their first Premier League victory in nearly a month. If they play as well as they did in the first half at United last month there is a possibility Eden Hazard or Willian may find something to smile about.
The Belgian openly questioned Conte's tactics as he felt that he was being played out of position and the addition of a striker such as Morata or Giroud would have made a huge difference in the result.
Giroud then hit the post after 66 minutes with a first-time shot from a low cross by Alonso.
Three points ensure that Chelsea stay in the hunt for an automatic Champions League qualification berth. The sense remains that this team boast more talent than those around them, together with an experienced coach very capable of organising them.More news: CSA suits in hot water over Sonny Bill masks
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Chelsea were on top for most of the first half and attempted a total of 16 shots, though both of their goals had more than a hint of fortune about them.
Chelsea were already leading 1-0 trough a Willian goal when Davide Zappacosta beat Wayne Hennessey with a shot. The Eagles are in 18th place, only a point from safety, but have a relatively easy run in, with only two games, including Saturday's, against the big six. They toyed with Palace at times and but for a late goal from their former defender Patrick van Aanholt, it would have been a comfortable evening.
Zaha caught Andreas Christensen dallying in his own half and played in Alexander Sørloth, one on one with Courtois.
These are just a few of the thoughts that will be whirring around in Conte's head as he finalises his game plan for Wednesday, and the choice he faces is the most important one he will make this season.
Meanwhile, Crystal Palace will be looking for a win to move them out of the relegation zone and have already beaten Conte's side this season as well as the corresponding fixture last term.
Subs not used: Caballero, Ampadu, Emerson, Moses.