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Sergio Aguero had levelled the score on 26 minutes but the visitors could not salvage a point after the hosts restored their lead with a rebound from the spot.
“Obviously, we are not happy but I can't say I'm disappointed with the performance,” he declared. “We are disappointed we couldn’t win the game.
"Other than that, it was a good performance. Physically, we stood up to the challenge and we dominated the game.
"You have to give credit to Chelsea - they defended well, they are solid and they really don't need a lot of chances to score a goal.
“I really have to say: we didn't look shaky. We were solid, we were dealing with the counter-attacks really well.
“Hazard and Pedro hardly ever got space to run away but he enters the box, creates something out of nothing and then they have a penalty.
"It can happen.”
Kompany was unfortunate to have a hand in the opening goal, deflecting Hazard’s first-time effort past Willy Caballero.
The Belgian admitted it was a stroke of bad luck.
He stated: "It's one of those: you hesitate and you don't know where to go.
“You expect the goalkeeper to be somewhere behind you obviously but if you let it go, people will say you should have headed it and if you head it, it goes so fast.
“Maybe if I had a little more hair, it would have gone further!”
Asked of City’s hopes for the season as a result of the defeat, Kompany replied: “Regarding our position, it will be: fight until the end for us. Out of something, there is always a new target.
"Our target is Champions League and hopefully, getting to the final of the FA Cup.”
A positive from the clash saw the skipper complete 90 minutes in a Premier League game for the first time since April 2016.
Reflecting on his own display, he expressed his relief and offered words of wisdom based on his experiences.
He continued: "It wasn't the easiest game to get back into but what I would say in terms of advice to younger players is: I could have stood back and moaned and complained that I wasn't playing but I never did.
"I worked very hard for that one opportunity and within the conditions that we available to me, I was helpful to the team.
“That's what I wanted to do."
The skipper is one of two changes to the side that drew at Arsenal on Sunday, with Fabian Delph also back in the starting line up.
Including TV details, team news and pre-match news and views from Pep Guardiola and the squad.
Dennis Tueart, Craig Bellamy and Carlos Tevez have all featured in classic clashes between Chelsea and Manchester City.