Diego Costa broke the deadlock for the hosts in the 35th minute before David Faupala levelled two minutes later for City’s young side.
However, Brazil international Willian restored Chelsea’s lead just three minutes after the break before Gary Cahill added a third at Stamford Bridge.
Eden Hazard scored Chelsea’s fourth 23 minutes from time before Bertrand Traore sealed the west London side’s passage into the FA Cup quarter-finals with a late goal.
Terry missed the FA Cup tie due to injury but the Chelsea centre-half was quick to praise his team-mates for their performance in his absence.
Terry wrote: “Great win in the F.A Cup today. Everton away in the next round @chelseafc. Supporters were different class 👏”
The 35-year-old centre-half sustained a hamstring injury during Chelsea’s win against Newcastle which forced the defender to miss their 2-1 loss to Paris Saint-Germain in the first leg of their Champions League last-16 tie on Tuesday.
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