Costa headed home Cesar Azpilicueta's cross in the 43rd minute at a foggy Selhurst Park to send Antonio Conte's men nine points clear.
"The run is fantastic," said Chelsea manager Conte. "Congratulations to my players, not me. I am pleased for them because they deserve this."
Zlatan Ibrahimovic scored twice as Manchester United won 2-0 at West Bromwich Albion, while champions Leicester City weathered the dismissal of Jamie Vardy to come from 2-0 down and draw 2-2 at Stoke City.
Chelsea have matched their record of 11 consecutive wins from April to September 2009, achieving the feat within the same season for the first time.
The only blemishes were yellow cards for Costa and N'Golo Kante that mean both players will be suspended for next weekend's home game against Bournemouth.
Narrowly beaten 2-1 by Manchester United in mid-week, Palace threatened to take the lead against Chelsea when Jason Puncheon scuffed wide from Martin Kelly's cross.
The visitors went in front shortly before half-time after Eden Hazard sprayed a pass wide to Azpilicueta, whose floated cross allowed Costa to nod in his 13th goal of the campaign.
Marcos Alonso struck the bar with a late free-kick, but Costa's goal proved sufficient.
Chelsea are nine points clear of second-place Liverpool, who visit Everton on Monday, and Arsenal, who travel to fourth-place Manchester City on Sunday.
"We can improve though," Conte added. "It is important to understand we are only at the 17th game in the season. We must not look at the table."
Jose Mourinho's United drew level on points with fifth-place Tottenham Hotspur, who host Burnley on Sunday, after winning at West Brom to record three successive league wins for the first time since August.
Ibrahimovic opened the scoring in the fifth minute with a header from Jesse Lingard's fine right-wing cross at the conclusion of a magnificent team move.
After Ben Foster had pushed a shot from Wayne Rooney onto the bar, Ibrahimovic added a second in the 56th minute.
- Negredo at the double -
The former Paris Saint-Germain striker darted between two defenders before scoring via a deflection off Craig Dawson, leaving him with nine goals from his last 10 appearances and 16 for the season.
"I keep saying that for me it is not his body, for me it is more up here," Mourinho told BT Sport, pointing to his head.
"It is a pleasure to see him play, his happiness. It's fantastic for a 35-year-old guy."
He added: "That was very good. We played really well at Crystal Palace and today, even better."
Leicester avoided a ninth defeat of the campaign thanks to an impressive fightback in a stormy game at Stoke, which left Claudio Ranieri's men three points above the relegation zone in 15th place.
Vardy was shown a straight red card in the 28th minute for a two-footed challenge on Mame Biram Diouf, ruling the England striker out of Leicester's entire festive programme.
Bojan Krkic put Stoke ahead with a 39th-minute penalty after Danny Simpson was adjudged to have handled his cross, with Joe Allen tapping in a second in first-half stoppage time.
But Leonardo Ulloa halved the deficit, smuggling a header over the line from fellow substitute Demarai Gray's cross, before Christian Fuchs teed up Daniel Amartey to head in an 88th-minute leveller.
Discussing whether Vardy deserved to be sent off, Leicester manager Ranieri said: "One manager says yes, one says no. If you listen to me, I say 'no'. If you listen to Mark Hughes, he says 'yes'.
"I don't know if he touched the player, but he touched the ball. I was angry (with the referee) because I saw my players fight and there were so many yellow cards, but it was a normal battle in the Premier League.
"It could be a turning point today, but we have to wait and see the next matches."
Middlesbrough moved above Leicester to 14th place after beating Swansea City 3-0 courtesy of a first-half Alvaro Negredo brace and a Marten de Roon volley.
Swansea dropped to second-bottom, with Sunderland climbing off the foot of the table after Patrick van Aanholt's 49th-minute goal secured a 1-0 home win over Watford.
Hull City are now bottom following a 1-0 defeat at West Ham United, who prevailed through Mark Noble's 76th-minute penalty after the visitors had hit the post three times.
source: news.abs-cbn.com
