Demba Ba's 28th-minute goal saw Chelsea return to third place above Tottenham in the Barclays Premier League for 24 hours at least, ahead of Sunday's north London derby.
West Brom had not won a league match at Chelsea in almost 35 years and although owner Roman Abramovich may be content with Benitez in charge until the end of the season and a fifth win in seven games, that may not be enough to appease Blues fans.
Many wondered if this would be Benitez's last stand following his rant against the title of 'interim manager' and the fans' incessant barracking of him. But instead he was able to savour a win, albeit a narrow one, to keep Chelsea firmly in the hunt for a Champions League place.
They created plenty of chances but Ben Foster saved well from Oscar while Juan Mata volleyed wide.
Oscar should have had a rare headed goal when he met Mata's clipped cross to the back post after 16 minutes but could not convert. Ba, Oscar and Hazard all had further chances before they took the lead from a corner.
The left-sided set-piece was played short to Oscar, whose inswinging cross towards the back post found David Luiz who headed back across goal for Ba to sweep home.
Further chances came for Chelsea and Frank Lampard, seeking his 200th Chelsea goal, had an effort ruled out early in the second half, with Mata flagged offside in the build-up.
Chelsea then appealed for a penalty when Hazard went down under pressure from Jonas Olsson, but referee Kevin Friend was unmoved.
With only a one-goal advantage, Chelsea had some nervous moments late on but they hung on for the win.