Oscar opened the scoring early on only for Anthony Pilkington to level with a little over 20 minutes to go.
But Chelsea struck twice in the last 10 minutes, Eden Hazard scoring from a breakaway following a Norwich corner and fellow substitute Willian adding the third on his Barclays Premier League debut.
But Cech admitted Chelsea did not have it all their own way and could have conceded a second when Norwich came back into the match in the second half.
Cech told Sky Sports 1: "They got an equaliser and it became much more difficult for us but the reaction was great.
"From the moment their equaliser went in we moved up a gear."
Cech added: "We had to work hard and there were moments in the game where everybody worked hard no matter how talented they are.
"They had some counter-attacks before we got the second goal. They didn't take their chances and we punished them for that."
Cech feels Mourinho's side are starting to show their real form but still sees room for improvement.
"We hope to get better every game and today was an important win for us. We hope we can take another step forward."
Man of the match Oscar added: "I am very happy with my goal. Today was a difficult game but Chelsea played very well. To win was very important for us."