The 33-year-old striker's 23rd-minute spot kick helped the Ivory Coast international join Stamford Bridge greats Peter Osgood and Roy Bentley on 150 goals.
Drogba scored from the penalty spot where the late Osgood's ashes are buried and after converting his kick he pointed to the spot, waved at the crowd and blew a kiss.
"It was an emotional goal for Didier and might be his last home appearance before next month's Africa Cup of Nations," manager Andre Villas-Boas told reporters.
"We will try our hardest for him to be available for the Portsmouth game (in the FA Cup on January 8)."
Before the match with Villa, the superstitious Drogba said his 150th goal could be a late "Christmas present for the fans" but defeat left Chelsea supporters fuming, with jeers echoing around Stamford Bridge after the final whistle.
Chelsea's title hopes now seem over after they dropped to fifth in the table, 11 points adrift of leaders Manchester City.
Drogba's future at Stamford Bridge is also uncertain after Villas-Boas said talks over a contract extension had not been resolved.
He has twice been voted Africa's player of the year.
