The Ivory Coast striker showed he was still committed to the Chelsea cause when he put the London side ahead after only seven minutes.
Chelsea played with nonchalant ease from the kick-off and it was no surprise when they went ahead with Drogba stroking the ball past Mark Schwarzer.
The visitors continued to look the better side with a superior brand of slick passing.
Boro struggled to get into the game in the early stages, but as play progressed they showed more determination with Gary O'Neil prominent in a number of attacks.
Middlesbrough centre-half Jonathan Woodgate was lucky to avoid a booking following a harsh tackle on Joe Cole. From the free-kick on the edge of the penalty area Drogba shot into the sidenetting.
Luke Young headed over the bar for Middlesbrough while Petr Cech made a good close-range save from Mido.
However, the game had not lived up to its billing with few thrills for the disappointing crowd of 27,699.
It was obvious that Middlesbrough would have to show more threat in attack if they were to pull this game round. So far the Chelsea defence had not been really stretched.
Middlesbrough skipper George Boateng, who was withdrawn later in the game, was booked for a foul, while Drogba was booed by the crowd while he lay on the ground injured. The referee helped Drogba to his feet.
Middlesbrough had penalty appeals turned down, with the decision to wave on play by referee Mark Halsey was a correct one.
After 57 minutes Chelsea increased their lead with a memorable first league goal by Brazilian international Alex.
What a goal to mark the occasion. Frank Lampard tapped the ball from a disputed free-kick to Alex and the Brazilian sent in a stunning shot from 40 yards.
It was a goal of tremendous power and accuracy and understandably Alex was mobbed by his team-mates. It is certain to be a goal-of-the-month candidate.
The game looked beyond Middlesbrough at this stage, for they had not seriously troubled the visitors.
In the match programme Middlesbrough manager Gareth Southgate made a plea for calm, writing: "There is no panic here about our league position or the number of points we have on the board.
"We have the opportunity to build as a unit and come together as a team. The only way to achieve that is through hard work on the training ground.'
Yet, results must improve if Middlesbrough are to avoid a relegation dogfight. The major worry is over who will score the goals.