The Ivory Coast international collapsed in agony during the goalless Champions League draw in Romania against CFR Cluj after lunging for a cross and jarring his studs into the turf.
Drogba was taken to hospital and there were fears he had suffered a serious knee injury but Chelsea said that he would not need surgery.
"Didier Drogba has had two forceful rotations in his knee and ankle. Fortunately, he has avoided any significant ligamentus rupture," a statement on Chelsea's website said.
"He will not be requiring surgery. We will keep him under observation for the next two or three days before putting an estimate on how long he'll be out, but we're certainly looking at weeks, not months."
The news will be a major relief for coach Luiz Felipe Scolari, who has been wondering how many players he would have fit for this weekend's Premier League match against Aston Villa at Stamford Bridge.
He is already without midfield dynamo Michael Essien who has a cruciate knee ligament injury, while Deco, Joe Cole and Ricardo Carvalho are also unfit.
Left back Ashley Cole missed Wednesday's match with a lower back ache while captain John Terry and fellow defender Alex also suffered a knock during the match.