Hazard set the Blues on the way to a first home league win of 2018 by shooting home after playing a one-two with Olivier Giroud, making his first start since signing from Arsenal.
Jay Rodriguez, Salomon Rondon and Jonny Evans all wasted chances for the Baggies before Chelsea sealed victory.
Victor Moses fired home the second before visiting goalkeeper Ben Foster was again beaten by Hazard.
The timely win lifts Chelsea back above Tottenham into fourth in the Premier League, three points behind second-placed Manchester United.
West Brom, who lost former Blues striker Daniel Sturridge to injury after just three minutes, are seven points from safety with 11 matches left. They have won just one of their past 25 league games.
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