Chelsea were left disappointed but still extended their lead to 10 points at the top of the Premier League after Burnley held them 1-1 on Sunday.
Champions Leicester City, however, sank deeper into trouble at the bottom, as a 2-0 defeat away to Swansea City left them a point above the relegation zone.
Manager Claudio Ranieri agreed that the Welsh club, who moved three points above his team, had been more determined and said that "it is not possible to continue this way".
Leicester remain in danger of becoming the first English champions to go down the year after winning the title since Manchester City in 1938.
The current City side under Pep Guardiola have an opportunity to cut Chelsea's lead back to eight points if they win at Bournemouth on Monday.
In an entertaining game at Burnley played amid flurries of sleet and snow, Pedro gave Chelsea an early lead from a pass by Victor Moses but the home side fought back well and equalized midway through the first half with a superb curling free kick by Robbie Brady, making his full debut.
Burnley, promoted last season, moved up one place to 12th and fully deserved a draw. They have earned 29 of their 30 points this season at home.
They restricted the leaders to only two shots on target, including the goal, which led Chelsea manager Antonio Conte to conclude his team could have done better with their 71 percent possession.
Reward for bravery
"We must be disappointed with the point because I think this type of game we could win," Conte said.
"We started very well and created chances to score. We could have killed the game but it didn't happen.
"Now it is important to restart."
Brady's equalizer was believed to be the first goal Chelsea had conceded from a direct free kick in four years.
"The lads were excellent and it shows what they're about," he told Sky Sports.
Manager Sean Dyche said the result was a reward for bravery.
"The mentality is the main thing I'm pleased with," he said.
"We wanted to do what we do. For the first 20 minutes we couldn't get it right, but I was super-impressed with the reaction to their goal."
Leicester's poor run continued and they have now not won in 15 away games in the league since last April or scored in six league games in 2017.
Ranieri, who was given a vote of confidence in an official club statement last week, saw his team go two goals down to Swansea before halftime to defenders Alfie Mawson and Martin Olsson.
They improved a little after he made two substitutions for the second half but Lukasz Fabianski still had little to do except make one save from Algerian striker Islam Slimani.
Leicester now have a break for an FA Cup tie at Millwall and then a Champions League tie with Sevilla before league games at home to Liverpool and relegation rivals Hull City.
"I wanted the start of a 'new season' today, but it is the same, it is unbelievable," Ranieri said.
"We have to speak together to find the solutions, it is not possible to continue in this way.
"The machine is not broken but when you don't play at a maximum level it is a difficult battle. They were more determined than us, that's the truth."
Swansea have now won three of their last four games under former Paris St. Germain, Real Madrid and Bayern Munich assistant manager Paul Clement.
"We have worked hard on the training ground to get the basics right, nothing complicated," he said.
Premier League: Chelsea extend lead, champions Leicester lose again