The Ivory Coast hitman scored the opener and ran the West Brom defence ragged before being substituted on 66 minutes.
Frank Lampard scored the other goal which dispatched relegation-haunted West Bromwich Albion with consummate ease at Stamford Bridge.
Scolari installed Drogba, Nicolas Anelka and Joe Cole as a three-man attack and the plan worked as the early goal at home, which has eluded Chelsea of late, came for the Blues on three minutes.
Drogba pounced on a deep right-wing cross by Joe Cole to plant a powerful header from eight yards out across the path of Albion goalkeeper Scott Carson and into the far corner of the net.
That should have opened the floodgates but the visitors dug in deep.
But it took until the 15th minute for them to have their first effort on goal - a Kim Do-Heon left-foot strike from just outside the Chelsea penalty area on the left. It was on target, but easily held by Petr Cech.
A fine three-man combination on 20 minutes involving Ashley Cole, Anelka and finally Lampard saw the midfielder ghost into the opponents penalty area and unleash a first-time left-foot shot, but sadly for the Chelsea skipper the ball sailed high and wide into the Shed stand.
Michael Ballack forced a smart save from Carson on 33 minutes when his long-range right footer fizzed across the Stamford Bridge turf.
Ashley Cole almost doubled the scoring on 35 minutes when he escaped his defensive marker Gianni Zuiverloon and let fly with a vicious left footer. Carson parried the ball onto the woodwork before seeing the ball bounce back to the left-back but this time his right footer was scuffed to Carson's grateful grasp.
The second goal came on the stroke of half-time and it was pure class. Joe Cole teased Albion on the left before sending a deft throughball to Ashley Cole. The ball deflected off the toe of substitute Leon Barnett to Lampard who chipped the ball delicately over Carson from eight yards out.
Scolari replaced Jose Bosingwa with Juliano Belletti at half-time.
Chris Brunt had a fine left-foot strike on the Chelsea goal on 56 minutes. The 20-yarder swerved in the air, but Cech was alert and positioned himself well to save easily.
Lampard then tried to repeat his first goal by chipping Carson on 59 minutes, but this time the goalkeeper stayed on his feet and plucked the ball easily out of the air.
Drogba almost scored his second on 65 minutes, blasting a rasping right footer which Carson stopped well. He was then substituted and with it Chelsea's most potent attacker and the match was effectively over.
Chelsea ran out easy winners and went top of the Premier League for a couple of hours at least while the Baggies can only look in desperation at the trap-door opening ever so slowly on their stay in the top flight.