Antonio Conte watched on as Hazard continued his return to full fitness after an ankle injury, showing signs of his dribbling ability.
Everton took the lead after 18 minutes when Joseph Colley turned Liam Walsh's cross into his own net.
After the break Walsh was again the creator, setting up Beni Baningime to score from the edge of the area, before Oumar Niasse made it 3-0.
Chelsea next face Sunderland on 9 September, with Everton playing Tottenham Hotspur two days later.
See: PL2 D1 table