The Portuguese defender, making his 150th appearance for the club, rose highest to meet Bridge's free-kick from the left and guided the ball past the outstretched Schwarzer for his first goal of the season.
It could have been even worse for Boro moments later, but Drogba could only plant his near-post header wide from Saloman Kalou's corner.
After their bright start, however, Chelsea took their foot off the gas and allowed the visitors back into the game.
Despite that, Boro struggled to create many clear-cut chances and Stewart Downing's low shot from distance, which Carlo Cudicini saved easily down to his right, was about all they good muster in the opening half.
Chelsea could have extended their lead as the half wore on but Kalou saw his shot blocked after good work from Michael Essien, and Joe Cole shot straight at Schwarzer after a marauding run from Drogba.
And just before half-time the Ivory Coast striker had a great chance to make it 2-0 but he headed Cole's cross over the bar when unmarked.
Boro substitute Alves was twice denied by the woodwork
The second half followed the same pattern as the first with Chelsea making most of the early running.
John Terry headed straight at Schwarzer and Essien's dipping volley flew just over, but the home fans were frustrated as their side's football lacked pace and penetration.
Manager Avram Grant replaced Michael Ballack with Shaun Wright-Phillips and he should have put the game beyond doubt as he first shot wide and then over when well placed.
Brazilian striker Alves should have levelled the score with 20 minutes remaining, when he beat the on-rushing Cudicini to the ball 35 yards out but hit the post with his shot.
And although the visitors swarmed forward in the closing stages, luck was on Chelsea's side.
First Alves headed onto the crossbar and Wheater followed suit with the rebound, before the ball fell to Jeremie Aliadiere who smashed his effort well wide.
Chelsea manager Avram Grant: "It was important to get the win at this stage of the season.
"There are a few games to go. It was not a great performance but it was very important that we got the points.
"Middlesbrough's set-pieces troubled us in the second half but I think we deserved the victory even if it was not a quality game.
"We didn't play as well as we can and sometimes this happens. But we created chances and scored a goal so that is what matters."
Middlesbrough manager Gareth Southgate: "I think the second half we were much better. In the first half we looked like we came here to swap shirts and watch Chelsea play.
"But in the second we rattled them, put pressure on Chelsea and I was pleased with the second half. We could have had a penalty and pushed them all the way.
"We changed the shape at half-time and our mentality changed as well. We seemed content to be losing 1-0 and needed to be more positive and test them which we did.
"This is a young group of lads and this kind of performance, especially the second half, will help them grow in belief."
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