At the other end of the table Swansea City improved their prospects of avoiding relegation with a 2-1 win at Everton in new Italian head coach Francesco Guidolin's first match.
Arsenal's German defender Per Mertesacker was sent off after 18 minutes for denying a goalscoring opportunity to Diego Costa, who scored the only goal five minutes later from a cross by Branislav Ivanovic.
The result preserved the defending champions' unbeaten record under experienced Dutchman Guus Hiddink, who took over as interim manager last month following Jose Mourinho's sacking.
They remain a bogey side to Arsenal, who have not beaten them in the last nine league games, failing to score in the last six.
Arsene Wenger's side were left in third place, level on points with Manchester City but three behind leaders Leicester City, who beat Stoke City 3-0 on Saturday.
Chelsea moved up to 13th, as high as they have been since beating Arsenal 2-0 at Stamford Bridge in September. On that occasion two Arsenal players were sent off -- although the red card to defender Gabriel was later rescinded.
Costa proved the key figure on Sunday before making way for substitute Loic Remy midway through the second half.
Arsenal, who took off striker Olivier Giroud after the sending-off in order to bring on Gabriel in defence, improved in the later stages, when Alexis Sanchez came on as a substitute, but they managed only one shot on target.
As Chelsea's captain John Terry claimed, his team were on top even before the sending off.
Terry insisted that reaching the top four to qualify for next season's Champions League was still not out of the question despite a 14-point gap with fourth-placed Tottenham Hotspur.
"Everyone's beating everyone and if we can put a run of games together, we are not writing anything off," he told Sky Sports.
Hiddink was more circumspect. "It's a big gap," he said. "But I want to see ambition always, whether you can (do it) or not.
"When you come here, you have to take the initiative and I'm proud of the guys."
Wenger felt Arsenal should have had a draw despite making so few chances but said they are still very much in the title race.
"It will be very tight until the end," he told Sky Sports.
Earlier Swansea's victory at Everton lifted them four points above the bottom three places.
A poor back pass by young Everton defender John Stones led to the 17th minute penalty from which Gylfi Sigurdsson gave Swansea the lead.
The home side soon equalised when Jack Cork touched Gareth Barry's flick into his own net, but a deflection off Stones from Andre Ayew's shot restored the Welsh club's lead before half-time.
They held out amid strong pressure in the second half, when Everton's leading scorer Romelu Lukaku twice went close and substitute Seamus Coleman twice missed badly, including from two yards out with the last kick of the match.
Achieving a second successive win for the first time this season put Guidolin's team in 15th place, and boosted their chances of avoiding relegation back to the Championship after five years.
"This is our target," Guidolin told Sky Sports. "These (two) results are very important for us."
The defeat left Everton 12th, with their former Swansea manager Roberto Martinez rueing a slow start and missed chances.
"The scoreline was very hard to take," he said.
(Reporting by Steve Tongue, editing by Pritha Sarkar)
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