Spanish forward Pedro put the visitors ahead in the fifth minute, hooking in a left-footed shot after being teed up by Cesc Fabregas.
Bournemouth, who shocked Chelsea 1-0 in the reverse fixture in December, were 2-0 down after 34 minutes when Hazard latched on to a Fabregas back-heel to end a drought going back 356 days.
Aguero treble sees Man City down Chelsea
The chants that welcomed the scoring return of last season's Player of the Year, with three games remaining, were cut short when Bournemouth hit back two minutes later through a Tommy Elphick header.
Willian made it 3-1 with a right-foot shot as Fabregas completed a hat-trick of assists and Hazard added his second from close range in stoppage-time.
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