Keane insisted after last weekend's 3-1 victory at Fulham that his side were not safe, and was not happy to have seen his caution proved well-founded.
He said: "We all need to do better, it's as simple as that. It is like when you win, you try not to over-analyse it and when you lose a game, you try not to over-analyse it. You learn from it.
"The way we started the game today, we have done that in the last few games. We need to start games well in the Premier League. Today was a long afternoon. I wouldn't say I enjoyed watching that today, and I am sure the supporters didn't. We need to do better, and that's including myself and the staff. We are all in it together.
"Whether people took their eye off the ball this week because everyone plugged into this rubbish that has been in the newspapers that we are safe and let's focus on next season - I have never heard so much rubbish in my entire life.
"We have got hard work ahead."
However, Darius Vassell, who had replaced fellow substitute Sturridge after he was injured in the penalty incident, snatched victory with a scuffed 87th-minute effort to make the home side pay for more poor defending.
Keane was unhappy with the penalty decision, but more concerned by his side's contribution to their own downfall.
He said: "At the time, we were never sure. It is easy for us afterwards to look at it, but it clearly wasn't a penalty. But there is no point complaining about it, is there? It's not going to change it."
Asked if he felt Sturridge had dived, he added: "It would be unfair of me to say that. A lot of players go down in the box and sometimes they lose their balance. There is no contact, but again, some players just lose their balance, that happens.
"It was a poor afternoon. The game wasn't great and we gave some bad goals away."
Delighted City boss Sven-Goran Eriksson admitted his side had ridden their luck, but was no less pleased because of it.
He said: "The penalty was a little bit harsh maybe - it was a contact, you can see that. I couldn't see it from the bench, but there was a little touch. But if you talk about luck, we still have to gain a lot this season.
"Our winning goal was not one of the greatest strikes I have seen in my life, but it went in and that's good. I hope we can keep this spirit up until the end of the season so we can at least still dream about playing in Europe next season.
"We still have a chance, although it will be difficult."