Chelsea hand Hodgson first defeat
Peter Odemwingie had West Brom ahead after 17 minutes but Drogba, Salomon Kalou and Frank Lampard turned it round by half-time as Chelsea bounced back well from their midweek Champions League exit at the hands of Manchester United.
Everton continued their hot streak with a 2-0 home win over Blackburn Rovers.
At the other end of the table Blackpool's dire run continued as a 3-1 home defeat by Wigan Athletic lifted the winners out of the bottom three and dragged Blackpool, who have won only once in their last 13, into it for the first time this season.
"I don't think I've seen a team this season keep Blackpool as quiet as we did, it was a great performance," Wigan boss Roberto Martinez told Sky Sports.
West Ham United are now second-last after conceding a late goal in a 2-1 home defeat by Aston Villa while Birmingham City beat Sunderland 2-0 in the day's other match.
After their Champions League exit, Chelsea's focus is now entirely on the Premier League and they now have Arsenal, and second place, firmly in their sights.
Manager Carlo Ancelotti, widely presumed by many in the British media to be on his way out of the club at the end of the season, started without Fernando Torres or Nicolas Anelka, his preferred strike duo at Old Trafford on Tuesday.
Drogba, who came off the bench to score in that match, took his starting opportunity with aplomb as he scored his side's opener and had a hand in the next two.
West Brom started full of the confidence that a seven-game unbeaten run builds and went ahead with a neat chipped finish by Peter Odemwingie after 17 minutes.
Drogba tapped in the equaliser five minutes later and when Scott Carson parried his 26th-minute shot, Kalou banged in the rebound.
Drogba then combined with Florent Malouda to set up Lampard for the third just before the break.
Chelsea cruised home after the break with Torres eventually joining the fray for the last seven minutes and even having the ball in the net - only for the goal to be ruled out for offside.
"It was a very important performance after the defeat at Manchester United," Ancelotti told Sky Sports.
"We were in focus, solid, played a good game and won. We reacted immediately (to their goal) and scored three in the first half."
The Italian said that Torres would feature more in the team's remaining games.
"I maintain my confidence in him," he said. "Maybe he's lost a bit of confidence but he has to know that everyone at the club has a lot of confidence in his quality."