Nemanja Matic says Chelsea wanted to win more than Arsenal, spurring the Barclays Premier League champions to a 2-0 victory.
Chelsea claimed their first home league triumph of the season and extended the Gunners' winless league run against their rivals to eight games.
Diego Costa was a nuisance throughout and could yet face Football Association scrutiny over his role in Gabriel's sending off.
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The FA will await referee Mike Dean's report before deciding whether or not the Chelsea striker has a case to answer.
"We wanted to win more than them," Matic told Chelsea TV.
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"It was a very important three points for us, especially against them.
"We didn't start the season very well so it was very important to win these two games - Champions League against Maccabi (Tel Aviv) and the derby against Arsenal.
"We didn't concede one goal and we scored six. We have to continue to play like this, especially the next game - it's going to be very important for us."
Chelsea next play Walsall in the Capital One Cup - a competition they won last term - before playing matches at Newcastle and Porto in the Premier League and Champions League, respectively.
Mourinho can be expected to make changes for the trip to the West Midlands, with John Terry likely to be restored in central defence.
Kurt Zouma was preferred to Terry to combat the pace of Theo Walcott and scored his first Premier League goal.
But the Frenchman knows he can learn plenty from 34-year-old Terry.
"I'm very happy because I had a chance to play some games and to show what I can do," the 20-year-old told Chelsea TV.
"I have to keep going. I have to learn a lot with the captain, the legend and Gary (Cahill). I'm very happy to play with them and learn a lot."
Mourinho is not looking beyond the Walsall match after a challenging start to the season.
"I want to go match after match," he said.
"Walsall is a cup game, a cup that we won and a cup that we don't want to throw away. We want to try to win. Wednesday is one more difficult match."
Another London derby awaits Arsenal, who visit Tottenham in the League Cup.
Aaron Ramsey told arsenal.com: "It's not good enough for a club like Arsenal to lose two games in a row.
"Hopefully now we can put things right on Wednesday. It will be a feisty game with hopefully some goals in our favour."
Goalkeeper Petr Cech spent 11 years at Stamford Bridge and was defeated on his return with the Gunners.
"You have to pick yourselves up and continue," Cech said.
"That's the only thing you can do because you don't win or lose the league after five games.
"You need to carry on until the last minute of the campaign. Obviously we need to keep going."