"We know they are a good side, everybody knows that, but we can't afford to play the way we did in the first half," said Hodgson.
"Both of the first two goals came about through balls lost near our own goal.
"I am very, very happy with the performance," said Tuchel. "It was clear we needed a complete performance in all aspects of the game. We did this, we were very brave with the ball and never lost intensity with the ball.
Sheffield United Chris Wilder speaking to BBC Sport: "Chelsea are flying it was a huge challenge in front of us as it has been all season and more recently playing Man City Man United and Chelsea - there's no time to get your head up we knew they'd have over 75%-80% possession thought shape was good.
Leicester boss Brendan Rodgers: "You have seen the real spirit of the side tonight. We were a real threat going forward.
"The performance and maturity of the team was pleasing. I knew we were playing against a talented team and we had to be right on our game.
"A month ago we were flying high, but since then we've struggled," Lampard told BBC Sport. "It's a challenge even a man down.
"Maybe a bit [confidence affecting his players], that's normal in football, but every team are having phases of form.
Chelsea boss Frank Lampard told BBC Sport: "We made mistakes, individual mistakes, and didn't get close enough. They punished us.
"If you want to reach the top level which City have been for a few years now, you have to accept the pressures.
Arsenal boss Mikel Arteta to BBC Sport: "The result is the main thing today, we really needed that win. We have been unlucky and frustrated with our results in the last eight weeks so this was a big day for us.
"It was a harsh scoreline but we were not good enough in our box and we weren't good enough in their box when we had the opportunities," Moyes said.
"Any small decisions did not go for us; the flow always went in Chelsea's favour tonight.
Wolves boss Nuno Espirito Santo to Amazon: "We were always in the game. We were well organised. In the second half we played really good football.
"In the first half the game was tight, Chelsea had the ball.
Lampard said it provided a dose of reality to those who say his side can win the league.
"It is a long season and these people talking about us winning the league with the best squad, it is ridiculous," he said.