Two pile-driving free-kicks by the Ivorian were sandwiched between an own goal by Abel Xavier.
John Terry returned to the Blues fold following his 14-match lay-off, but Boro's England centre-back Jonathan Woodgate missed the match with a hamstring injury suffered playing for England in the midweek defeat to Spain.
Drogba had the first attack on goal on seven minutes, when he latched on to an inch-perfect through ball by Frank Lampard but the striker blasted too high from just inside the penalty area.
A minute later and Drogba flashed a left-footer inches wide of Mark Schwarzer's right post.
But Boro should have gone into the lead with their first attack. Mark Viduka missed a sitter on 12 minutes from six yards when he somehow managed to head wide of Petr Cech's goal from a brilliant Stewart Downing left-wing cross.
Middlesbrough survived an attack on goal on 20 minutes when Lassana Diarra went on a mazy run along the edge of the penalty area. He let fly with a shot but Emmanuel Pogatetz beat it down then Xavier prevented Andriy Shevchenko from getting through.
Chelsea boss Jose Mourinho's frustration spilled over on to the touchline on 33 minutes when he and Boro boss Gareth Southgate squared up to each other following a challenge between Pogatetz and Drogba. The fourth official calmed both men down.
But Chelsea took the lead on the stroke of half-time through Drogba's wonderful free-kick from just outside the penalty area.
Chelsea started the second half as the first and could have increased their lead on 48 minutes, but Julio Arca cleared a fine Shevchenko header off the line following a Lampard corner.
Chelsea had a let-off when Boro hit the woodwork as Andrew Taylor sent in a hopeful left-wing cross on 50 minutes. The ball swerved towards the top right-hand corner of the net but Cech managed to get a touch to it and the ball ricocheted off the angle of crossbar and right post.
Middlesbrough's counter-attacking policy almost struck gold on the hour when the ball was fed to Yakubu, who went round Paulo Ferreira and then Claude Makelele before letting fly from just outside the area. Luckily for the Blues the left-footer was high and wide of Cech's goal.
Arjen Robben had been on the pitch for a mere 30 seconds on 65 minutes before he went past Downing and Taylor on the right before falling in the penalty area.
However he got up to produce a one-two with Salomon Kalou before sending in a cross which was deflected past Schwarzer by the hapless Xavier for an own goal.
Schwarzer was called into action ten minutes from time when he acrobatically tipped away a fierce Drogba drive after a woeful clearance by Xavier.
Goal number three came three minutes later - again from the boot of Drogba.
Wide on the left, the striker sent over a powerful right-footed curler of a free-kick, which deflected off the head of Andrew Davies in the wall and sailed past the despairing dive of Schwarzer and into the net.