Successive three-goal defeats had seen the pressure grow on Blues boss Antonio Conte, but his side dug deep to pick up just a fourth victory in 11 attempts this calendar year.
Albion, boosted by the return of three previously injured players to their starting lineup, started the match brightly and had a strong call for a penalty turned down inside the opening 10 minutes.
Cesc Fabregas, brought in for the suspended Tiemoue Bakayoko, lunged in on Jay Rodriguez - an early sub due to yet another Daniel Sturridge injury setback - and was fortunate to avoid conceding a spot kick and being dismissed.
Rodriguez spurned a great opportunity moments later when drilling the ball wide from the edge of the box, despite having time to carry the ball closer to goal following a poor Moses header.
Out-of-form Chelsea, unbeaten in 13 home matches against West Brom, slowly started to grow into the match and were good value for their goal when it arrived 25 minutes in.
Giroud was heavily involved in the move on his first start for the club, holding off Ahmed Hegazi in the box and feeding a return pass into the path of Hazard to sweep the ball past Foster into the bottom corner.
Albion saw more of the ball between Chelsea's opener and half time, but it was the Blues who were looking the more lively with some decent interplay in the final third.
Chelsea were having to hold on to their lead, with Rondon getting the better of fit-again Andreas Christensen shortly before the hour, only for his one-on-one to hit Thibaut Courtois and fly into the side-netting.
Jonny Evans got his head to the resulting corner, sending the ball spinning narrowly over the bar, but the missed chances would prove costly as the home side soon doubled their advantage.
Another give-and-go proved to be Albion's undoing, this time after Moses exchanged passes in the box - the final touch coming off Craig Dawson - and coolly placed the ball past Foster.
A third goal would arrive through some more smart work from Hazard who, moments after seeing Moses kept out by the leg of Foster, cut inside on his left and smashed the ball past the wrong-footed keeper.
Alvaro Morata was brought off the bench for his return to action and forced Foster into another save inside the final 10 minutes, proving to be the final piece of meaningful action on a happier night for Conte and his players.
CHELSEA (3-4-3): Courtois; Azpilicueta, Christensen (Cahill 74'), Rudiger; Moses, Kante, Fabregas, Zappacosta; Pedro (Willian 80'), Giroud (Morata 61'), Hazard
WEST BROMWICH ALBION (4-4-2): Foster; Gibbs, Evans, Hegazi, Dawson; Brunt (Burke 46'), Barry (Yacob 80'), Krychowiak, Phillips; Sturridge (Rodriguez 4'), Rondon