Heavy back-to-back defeats against Bournemouth and Watford led to evermore voluble questions surrounding the future of head coach Antonio Conte and placed the reigning champions' hopes of a top-four finish in jeopardy.
Captain Gary Cahill dropped to the bench but the error-strewn element of recent outings was again in evidence and Salomon Rondon and Jay Rodriguez both could have put Albion ahead before Hazard did so for Chelsea.
There was further defensive generosity to enrage Conte until Victor Moses doubled the Blues' advantage after the hour and a sublime 15th of the campaign from Hazard made the points safe.
7 - Eden Hazard has scored more Premier League goals versus West Brom than he has against any other team in the competition (7). Thorn. pic.twitter.com/XB1bKzPek6— OptaJoe (@OptaJoe) February 12, 2018
Daniel Sturridge was making his second start for West Brom but the on-loan Liverpool striker's injury jinx struck again, a hamstring injury forcing him off in the fourth minute.
Replacement Rodriguez almost had an instant impact, thumping wide before Chelsea midfielder Cesc Fabregas crashed into him. Referee Lee Mason was not interested in his penalty claims.
Salomon Rondon then headed off target from Matt Phillips' brilliant, fizzed cross and Rodriguez dragged wastefully wide after Moses' woeful backwards header left him through on goal.
Chelsea hauled themselves out of an abject slumber as Davide Zappacosta and full-debutant Olivier Giroud forced close-range saves from Ben Foster and they led in the 25th minute.
Hazard effortlessly shimmied around Gareth Barry and fed Giroud for precisely the type of centre-forward play Conte sought during the January transfer window.
The ex-Arsenal man spun Ahmed Hegazi and completed a one-touch lay-off for Hazard to thrash into the bottom corner.
Giroud required a large head bandage after taking an accidental boot to the head from Hegazi, nodded narrowly wide from Pedro's dinked cross and limped gingerly to the touchline after being clattered by Jonny Evans. He was probably glad to hear the half-time whistle.
Rondon should have equalised in the 57th minute when he outmuscled a dithering Andreas Christensen but failed to beat Thibaut Courtois. From the resulting corner an unmarked Evans headed over.
A battered and bruised Giroud took his applause from an appreciative Stamford Bridge crowd after the hour, with Alvaro Morata introduced up front, and, as in the first half, Chelsea gathered themselves to punish West Brom's wastefulness.
Fabregas looked for Morata with a dinked pass and it fell kindly for Moses to rifle beyond Foster.
Moses cut inside seeking a second, only to be denied by the West Brom goalkeeper's boot before Hazard showed him how it was done in the 71st minute – leaving Evans in his wake and unleashing an unerring left-footed finish.
It proved a harsh lesson in expert finishing for Albion, whose situation at the foot of the table and seven points from safety with 11 games to play is a desperate one.