Sturridge scored the opener for the Blues in a 3-1 win over Everton at Stamford Bridge which left them a point behind second-placed Manchester United and a further two behind the England Under-21 striker's old club Manchester City.
The Manchester clubs meet for the first time in the league this season next weekend and Sturridge, whose goal was followed by further efforts from John Terry and Ramires, does not mind all the attention staying on the north west's big two.
He told ESPN: "I'm happy for everyone not to talk about us, for us to get on with what we're doing. It doesn't matter whether people talk (about Chelsea) or not, we just want to continue beating every team that is put in front of us. It doesn't matter what the opposition is, we want to win the game."
Midfielder Juan Mata, who had another influential evening for his new club, added: "I enjoyed the game, it was important for us and we needed to win to keep up in the table."
Asked about his great start to life at Chelsea, the former Valencia man said: "In my first match I got a goal and it gave confidence to me. I am happy to be here, in the city and the club."
Everton had only a late consolation from Apostolos Vellios to show for their efforts, and David Moyes could not fault his side's application.
The Scot said: "We have probably had tougher games here against Chelsea but conceded less goals. We were disappointed with the goals we conceded but I give a lot of credit to the players, they kept at it right to the end.
"The second goal (by Terry close to half-time) took the wind out of us a little bit because it would have been different, we would still have been in the game at 1-0.
"It's been that way several times when we've been here before and you always have a chance but when it went to two it made it a mountain to climb, especially because they keep the ball well. We were hoping they would start protecting the 1-0 and we would get something back.
"The players showed a great spirit and at times that takes you a long way. We're just disappointed we're beginning to lose one or two goals that with a bit better defending we could have stopped."