The Blues fell 2-0 behind to the reigning Premier League champions at Wembley following Samir Nasri's first-half strike and a Sergio Aguero goal two minutes after the break.
Rafael Benitez's men threatened a comeback when Demba Ba scored in the second half but were unable to see it through.
Belgian international Hazard believes Chelsea showed City too much respect in the opening half and paid the price.
He told The Sun: "We were frightened of City in the first half.
"We showed them too much respect, and paid the price. Maybe we might have deserved a different fate, but we started the game badly.
"We played the final 40 minutes, but that is not enough to beat City.
"We had chances, and we believed we could do something, but their keeper made some good saves."
Defender Ashley Cole was absent from the Wembley clash with a hamstring injury suffered during the quarter-final win over Manchester United and manager Benitez has confirmed he will be missing again when Chelsea travel to neighbours Fulham in the Premier League on Wednesday.
Benitez said: "He will not be back for Wednesday. He is getting closer but not for this game."