It was a quiet first period from both sides, with a half opportunity from Jason Puncheon before Costa headed home in the 43rd minute. The second period however burst into life with the Blues dominating the proceedings as opportunities fell to N'Golo Kante, Victor Moses, Marco Alonso and Cesc Fabregas but they failed to add to the scoreline.
Making the crucial difference
Antonio Conte's side traveled to Selhurst Park in a rich vein of form looking for their 11th consecutive league win, but it was a slow start to the proceedings in the lunchtime kick-off.
The first sniff at goal from either side didn't arrive until the 21st minute, when Martin Kelly played the ball into the area for Puncheon but the FA Cup final hero scuffed his effort wide of the far post.
The next opportunity at goal brought around the game's only goal with two minutes to play. It was a simple finish from the visitors, as Cesar Azpilicueta played a cross in and Costa rose highest to head the ball back across Wayne Hennessey and into the corner of the net.
Twisting the screw
It seemed that Conte had reiterated the idea of pressure at the break, as his side came flying out of the traps and had numerous opportunities to double their lead. The first arrived in the 51st minute as Kante let one rip, his effort was swerving all over the shop but Hennessey did well to push the Frenchman's effort away from goal.
The resurgent Moses then had a crack at the Welshman's goal in the 63rd minute, he did well wiggle his way into the area and managed to get his effort away but Joel Ward did well to make the crucial block.
Moses then turned provider minutes later, as his cross into the area looking for the head of Costa was cleared by Scott Dann. However the skipper's attempted clearance only made it as far as Alonso, who decided to hit the effort on the volley but it could ripple the side-netting.
One last punt
Chelsea's bombardment on the Palace goal failed to subside as the clock ticked down, as they looked to cement another crucial three points.
Fresh from his heroics midweek, Fabregas received another opportunity in the 70th minute after Eden Hazard's cutback, but the effort was tame and straight at Hennessey.
Kante had another crack at it minutes later, and it looked like the Premier League winner had struck lucky with his low effort but Hennessey was there once again to block the effort.
Their best opportunity to see out the victory arose six minutes, Alonso set himself up for free-kick and the Defenders effort looked good as it went over the wall but it cannoned back off the crossbar.
The final opportunity fell to Alan Pardew's side deep into stoppage time, as Andros Townsend looked to replicate Alonso's efforts but the set piece went high over the crossbar.