Chelsea made hard work of a 2-1 win over relegation-threatened Crystal Palace after Fernando Torres' opener was cancelled out by Marouane Chamakh, with Ramires netting what proved to be the winner in the first half.
Mourinho's men are now within two points of leaders Arsenal, who lost at Manchester City earlier on Saturday, and can assume the Premier League lead by winning at the Emirates Stadium on December 23.
Mourinho, who watched most of the second half of the match at the Etihad Stadium prior to kick-off at Stamford Bridge, said: "I'm not surprised because it's our nature. It's in this club's DNA over the years to try to fight for trophies.
"It's my way of work, too; it's the mentality we want the players to have.
"We all know that this is a transitional period, we all know that we have just a few players from the winning side from the past. We all know we have lots of young talent to improve.
"I believe in another club with a different profile everybody would say 'okay, let's work and wait calmly, the results will come in the near future'.
"We are not like that. We want to build, but at the same time we want to fight for titles. And in this moment in the Premier League we are there, two points behind the leader in a group that is there."
Whether Chelsea do assume pole position by Christmas is not significant, according to the Portuguese.
"It's not important," Mourinho said.
"In this league to be top of the league by December or even by the end of January, I don't think is crucial. I think this will be open till very, very late.
"Every game is very hard. The distances (between clubs in the table) they open and they close, they open, they close.
"Of course we go there to try to win and we go there with the objective of being top of the league, but if we don't, I don't think it's crucial."
Mourinho recognised Palace could trouble his side, who continued to lack the ruthless streak City have demonstrated at home despite extending their own formidable record at Stamford Bridge. Chelsea are now unbeaten in 16 league matches at home, as Mourinho's record was extended to 68 undefeated.
The Portuguese added: "I was expecting an easy victory; I was expecting big problems. I was ready for both things.
"When I was reading the game and analysing the situation I was feeling the game could go in both directions.
"You score a goal that kills the opponent, 2-0 or 3-1, the game is over and you can go to a different result.
"When you don't do that, I knew they were a team who would give problems.
"For me, this result could be 5-1, could be 2-2."
Palace have now won two of Tony Pulis' four matches in charge to return to the pack after appearing likely to be cast adrift.
Pulis said: "To be in the game for 94 minutes is testament to the commitment and the effort.
"The important thing is that the lads keep believing that we have got a chance and that we keep playing with that spirit and commitment."
The former Stoke boss reserved special praise for the Palace goalscorer.
"I thought when we got it to Chamakh we played some good stuff," Pulis added.
"Chamakh slows it down for us, gets a pass off and gives us time to get runners up the pitch.
"Bolasie came on and was a threat, Puncheon was a threat and Cameron Jerome was smashing. There's a threat there."
Contrary to outside perceptions, Chamakh has applied himself well under Pulis.
The Welshman added: "The lads have bought into what we're trying to do. The most important thing is to galvanise a spirit within a football club.
"Marouane has bought into that. You've seen the quality, he's a top player. Wenger doesn't buy players who can't play. The kid can play.
"It's helped, I think, that he's been given the responsibility to play just off the striker, where he's got more room and space and he hasn't got his back to play all the time."