Belgian Hazard picked up the ball in his own half in the ninth minute and with a burst of speed wrong-footed defenders, cut inside Michael Keane and sent a right foot shot low past a diving keeper Tom Heaton.
Willian's contribution came in the 41st minute after an exchange of passes between Hazard and Spain striker Diego Costa. Costa sent the ball to the Brazilian who sidestepped Stephen Ward and drilled the ball into the net.
Heaton was called into action constantly in the second half as Chelsea peppered the Burnley goal with shots. But he could not keep out Moses, who got on the end of a ball from fellow-substitute Pedro in the dying minutes, ensuring a 100 percent start to the campaign for the London side.
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(Reporting by Clare Lovell, editing by Ed Osmond)
