The Ivory Coast striker's 26th minute left-footer from eight yards out was an outstanding finish from the best move of the game in a frankly disappointing performance from the champions-elect.
An excellent defence-splitting throughball by Frank Lampard found Damien Duff wide on the left.
The Irishman had judged his run superbly and his cut-back across goal was met by the Chelsea front man who drove home past a hapless Russell Hoult in the Baggies goal.
With nine games remaining Chelsea are odds-on to lift their first top-flight League title in 50 years but this won't go down as a Championship performance.
They outplayed West Brom, had the lion's share of possession, but poor finishing, in particular from Drogba, meant that their opponents were never out of the game.
In fact, they could claim to have scored a good goal from Geoff Horsfield but it was judged to be offside.
But it was a catalogue of misses from Chelsea which characterised the match. Drogba, on a better day, could easily have scored five goals
He was sent clear in the 32nd minute by a Joe Cole pass but, although he skipped past Hoult, desperate defending from Martin Albrechtsen stopped the striker slotting the ball home.
Drogba missed another sitter in the 71st minutes when Duff's left-wing cross left him free in the six-yard box, but he somehow conspired to head wide with the goal gaping.
Kanu had the best effort for the Baggies a minute later when a fine right-footed volley from the Nigerian arrowed towards goal from just inside the penalty area. However, the ball smacked into the face of defender Robert Huth.