Diego Costa headed in Hazard’s cross in the 35th minute in west London but City equalised 94 seconds later when David Faupala bundled home.
However, Willian’s drive restored Chelsea’s lead three minutes after half-time.
Gary Cahill fired in a third from 20 yards and Hazard curled in a free-kick for his second goal of the season, before Bertrand Traore’s header added further gloss to the scoreline.
Oscar also had a penalty saved by goalkeeper Willy Caballero as Chelsea scored five at home for the second weekend running.
Speaking to BBC Sport after the game, Belgium international Hazard said: “We did very well in the second half.
“The target is to win this FA Cup and if we want to win, we have to beat the best teams.
“OK, today they played with their second team but we did very well and we deserved to win.”
Chelsea, who were beaten 2-1 by Paris Saint-Germain in midweek, now turn their attentions towards their Premier League trip to Southampton on Saturday afternoon.
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