Didier Drogba's cool finish set Chelsea on their way to victory and increased the pressure on Gareth Southgate's struggling Middlesbrough side.
Drogba wheels away after putting Chelsea ahead at the Riverside
Drogba showed great composure to slot home after racing through following a slick one-two with Frank Lampard.
Mido missed a glorious chance to level moments later, Petr Cech reacting brilliantly to keep his header out.
But Chelsea sealed the win after the break when Alex crashed a wonderful 30-yard free-kick past Mark Schwarzer.
Up until Alex's stunning strike the game was in the balance but the second goal relaxed the visitors who suddenly started to play with a swagger and saw out the remaining half an hour in relative comfort.
We didn't get close to them until they scored their goal but maybe by then it's too late
It was Drogba, who this week went public with his desire to quit the club, who eased the early pressure on both himself and the club with the crucial opening goal.
Florent Malouda fed the ball into the Ivorian whose cushioned pass found Lampard.
The England midfielder then played a first-time pass through to Drogba who raced towards goal and coolly guided his shot across Mark Schwarzer and into the far corner.
But Boro, who had beaten Chelsea in the corresponding fixture for the past two seasons, should have levelled almost immediately.
Gary O'Neil skipped past Paulo Ferreira - playing out of position at left-back - and his teasing cross found Mido whose powerful close-range header forced a fine instinctive save from Cech.
After that the visitors seemed content to sit back and defend their lead and Boro had plenty of possession but the closest they came to the Chelsea goal were a couple of dangerous crosses which Juliano Belletti did well to clear.
At the other end, Chelsea's only noteworthy chance of an error-strewn first-half saw Drogba curl a free-kick wide from the edge of the box.
Boro upped the pace after the interval and looked threatening. Mido should have done better with a floated Stewart Downing cross as the home side enjoyed a decent spell camped in the Chelsea half.
But the game was effectively over when Alex spanked Chelsea 2-0 ahead with a thunderous free-kick on 57 minutes.
Lampard touched the ball to his right and the Brazilian smashed it high into the right-hand side of the net from 30-yards - his first goal for the club.
The Londoners then had a couple of half-chances to increase their lead through Malouda and Lampard.
O'Neil blazed a shot over the bar in the dying seconds after some dreadful defending but Boro never looked like avoiding a third consecutive Premier League.
Middlesbrough boss Gareth Southgate:
"Against a team with their quality you can't concede early and that's the problem.
"We didn't get close to them until they scored their goal but maybe by then it's too late.
"Petr Cech made a fantastic save (from Mido) we created two or three very good chances and are well in the game but the free-kick was top drawer and that killed the game off.
"When you have a spell like this you need to stick together and roll your sleeves up."
Chelsea manager Avram Grant on the win and Didier Drogba's comments in the French press that he wants to quit the club:
"We played good and we won. I'm happy with win and happy are playing better from game to game.
"We started well, scored and could have scored more. Alex has a terrific shot and it was a fantastic goal and the first goal was a good combination too.
"I have seen Didier in the last month in training, in my room, on the pitch and I haven't had one problem. If he has done something off the pitch and he regrets it, it's good.
"He is not a negative person. If he was negative on the pitch or in the dressing room or when he spoke to me, that is another story. But it's not like this. Players need to speak on the pitch, and he did that very well."
Middlesbrough: Schwarzer, Young, Woodgate, Riggott, Taylor, O'Neil, Boateng (Cattermole 67), Rochemback, Downing, Sanli (Craddock 84), Mido.
Subs Not Used: Turnbull, Lee, Wheater.
Chelsea: Cech, Belletti, Carvalho, Alex, Ferreira, Obi, Essien (Sidwell 81), Lampard, Joe Cole, Drogba (Shevchenko 85), Malouda (Wright-Phillips 75).
Subs Not Used: Cudicini, Ben-Haim.
Goals: Drogba 8, Alex 57.
Ref: Mark Halsey (Lancashire).