The Spain international climbed high to connect with Cesar Azpilicueta's cross in the 43rd minute, guiding a header beyond the reach of Wayne Hennessey at a foggy Selhurst Park.
It was Costa's 13th league goal of the season, taking his Chelsea haul to 50 in all competitions, and was enough to seal a third consecutive 1-0 win for Antonio Conte's confident side who head into Christmas looking unstoppable.
After 17 games they have 14 wins - a record for a new manager in the Premier League - and are nine points clear of Arsenal and Liverpool who play on Sunday and Monday against Manchester City and Everton respectively.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
It is the first time Chelsea have won 11 consecutive league games in a single season, although they also managed 11 in a run spanning two campaigns in 2009.
With home games to come against Bournemouth and Stoke City they are closing in on Arsenal's record of 14 consecutive wins in the Premier League.
"Another win, we deserved to win," Conte told Sky Sports. "We played with great concentration against a strong, physical team.
"To win 11 in a row is fantastic. The most important thing is that we are a team, in every moment of the every game. When we suffer we suffer together. I see every day their commitment."
Without being scintillating Chelsea were well worth the win although they offered precious little until Costa opened the scoring. The first-half action matched the gloomy weather but Chelsea went in ahead thanks to a goal of stunning simplicity.
Eden Hazard played the ball to Azpilicueta and his dinked cross was perfect for Costa to find the corner of the net.
They will miss his predatory instincts against Bournemouth on Boxing Day, however, after he picked up an early yellow card and a one-match suspension.
Palace were energetic at the start of the second half but could not ruffle a Chelsea defence in total command.
The visitors looked the more likely side to score again with Marcos Alonso firing into the side-netting.
N'Golo Kante forced a fine save from Hennessey before Alonso struck the underside of the crossbar with a free kick and Cahill was unable to nod home an inviting rebound.
Palace have won only once in their last 11 league games and remained three points above the bottom three.
(Reporting by Martyn Herman, editing by Ed Osmond)
