With Arsenal being held at home to Manchester City, Chelsea take a seven-point lead over the champions into 2005.
With Manchester United also dropping points at home to Spurs, this was a night for Chelsea to cement their position at the top of the table.
Drogba was well worth his man-of-the-match award for the beauty of his goals.
They came in a two-minute spell early in the first half which had started well for Middlesbrough.
However after weathering an opening barrage, Chelsea soon settled into their customary passing mode and after 15 minutes, Drogba hit his first.
A well-executed counter-attack saw Frank Lampard twice involved. His first-time pass to Arjen Robben was finished off with a slide-rule pass to Drogba.
The Ivory Coast striker turned centre-back Gareth Southgate inside out, resisted a shirt pull from the former England defender before drilling a low right-footer, from the edge of the penalty area, past goalkeeper Mark Schwarzer.
Question marks could be levelled against the Australian keeper for the second goal in terms of his positioning but the ferocity of Drogba's header was probably too much for Schwarzer.
Chelsea were rewarded with a free-kick 20 yards out in the 17th minute, just on the right apex of the area after Joe Cole had been upended.
Lampard sent in sizzling, curling right-footer which was met by the head of Drogba. He easily beat defender Colin Cooper and arrowed his header downwards and towards goal. Schwarzer may have misjudged the flight of the ball but the power was too much for the hapless shot-stopper.
The game was over as an event and only a Damien Duff rasper in the 58th minute stirred the Chelsea faithful into song.
Middlesbrough's best effort on goal came from the mercurial feet of Stewart Downing in the 66th minute when his low, right footer was on target but well held by Petr Cech.
In truth Middlesbrough were poor, bereft of ideas up front and more than happy to leave Stamford Bridge with their pride intact.