Willian gave Chelsea the lead in the 13th minute with a free-kick, only for the Potters to draw level through Jonathan Walters' penalty after the Stoke striker was pushed in the penalty area by Cahill.
Antonio Conte's men are now 13 points clear at the top of the table with another hard fixture ticked off the calendar.
"This win is important, honestly".
After a 3-1 win over Swansea City last month, Conte said he thought 29 points were required in order for the Blues to guarantee the return of the top-flight trophy to Stamford Bridge. First, we will try to win the next game in the league against Stoke and, when the cup comes, we are going to try to win it. "He is a type of player who is very hard to stop and for this reason you try to stop him with a good or bad way", Conte said. I hope to win it, but there is a long way to go.
"But I'm not anxious about this because Eden is very strong physically, and he's showed me he's able to play in this league".
"He showed me great commitment and that he thinks for the team and not for himself. I want to continue this way". "He was great, he played a hard game but he always showed danger for the defensive line for Stoke".
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Fortunately, Arsenal put on a stellar display during the following 45 minutes. But it was a football education for sure".
On what he felt of Stoke's approach towards Costa, Conte continued: "It's not easy to reply to this type of question".
Lee Grant in Stoke's goal had clearly looked out the Teflon gloves for today as a 45 YARD free kick from David Luiz nearly caught him unawares, bouncing off his chest.
Just as against West Ham, Chelsea scored with their first real attack but despite occasionally threatening a second, with Alonso stinging Grant's palms after latching onto a brilliant Victor Moses dink through the Stoke defence, Conte's side never appeared fully comfortable or in control. To score the winning goal is great for him and our team. You must know this.
Conte was without influential winger Eden Hazard, the club's second top goalscorer, but confirmed the Belgian's absence was only due to a minor issue.
"In the last training he felt a bit of a problem in his calf, for this reason I decided not to risk him".
Conte went on to reveal that he did have a couple of injury doubts in his squad but refused to name the players in question in Friday's press conference.