Boro arrived in West London knowing that they needed a victory to realistically stand any chance of bridging the seven-point gap to Swansea City, but they were utterly outclassed as strikes from Diego Costa and Nemanja Matic either side of a Brad Guzan own goal took Chelsea to within one win of the Premier League title.
The Blues now sit seven points clear of Tottenham Hotspur at the top of the table and will wrap up their second title in the space of three years with victory over West Bromwich Albion at The Hawthorns on Friday night.
Chelsea were without newly-crowned FWA Footballer of the Year N'Golo Kante due to a thigh injury, but Cesc Fabregas was not a bad replacement and the Spaniard went on to pull the strings during a dominant display from the home side.
Chelsea could have been ahead within two minutes when Fabregas floated a diagonal ball into the left channel for Marcos Alonso, who stole in unmarked at the back post before firing a sweetly-struck first-time volley towards goal which Guzan fortuitously deflected against the crossbar.
It was an avenue of attack that the hosts repeatedly exploited as Boro struggled to come up with an answer, and they were lucky to avoid being punished after 17 minutes when Fabregas placed an effort comfortably wide of the target after being picked out by Eden Hazard.
Steve Agnew's side were afforded another let-off shortly afterwards when Hazard again broke behind the defence down the left channel and played a teasing low pass across the face of goal which George Friend did just enough to prevent Costa turning home from close range.
Costa did not have to wait long for his 20th Premier League goal of the season, though, being picked out by another diagonal ball from Fabregas before taking a touch and slotting his finish through Guzan's legs from close range.
A second goal always seemed likely, and it was no surprise that it arrived via a similar means, with Cesar Azpilicueta this time playing the diagonal pass for Alonso, whose volley across the face of goal deflected in off the legs of Guzan.
Two almost became three shortly afterwards as Chelsea really began to drive home their dominance, with Fabregas spreading another pinpoint pass to release Victor Moses, whose low drive was kept out by the Middlesbrough keeper.
Alonso came close again with just four minutes remaining until half time with a deflected free kick that flew just over, but Chelsea went into the break in complete control and unfortunate not to be even further ahead.
If there was any rousing Boro teamtalk ahead of their final 45 minutes before relegation then Chelsea were quick to reassert their dominance upon the restart and, having hit the crossbar within two minutes of the first half, they did the same in the opening minute of the second.
It was a sign of things to come as Chelsea continued their onslaught throughout the second half, with Azpilicueta sending one 25-yard drive narrowly over before Costa was inches short of turning Alonso's cross/shot home from close range.
Gary Cahill was the next to let fly with a thunderous first-time effort as players throughout the Chelsea team tried their luck against their besieged visitors, but Guzan was able to beat the skipper's strike away at the near post.
Fabregas was close to helping himself to a deserved goal moments later when his volley came off Calum Chambers before trickling agonisingly past the post.
The third goal eventually arrived shortly after the hour mark as Fabregas clipped a pass in to Matic, who chested the ball down inside the area before firing his finish through Guzan's legs - completing a hat-trick of nutmegs for the Boro keeper.
It was also Fabregas's 10th assist of the season, making him the first player in Premier League history to reach double figures in six different campaigns.
Boro's slim hopes of survival were dead and buried by that point, but the visitors did finally draw a save from Thibaut Courtois with just over 20 minutes remaining when the Chelsea keeper collected Grant Leadbitter's low strike.
If either side were going to get a fourth goal then it would have been Chelsea, though, and Pedro blazed another opportunity over the crossbar before Guzan was called upon to deny a powerful strike from Moses.
Chelsea's dominance was such that the full-time whistle may have been a welcome sound for Boro, despite it confirming their relegation from the top flight for a record-equalling fourth time with two games to spare.
Boro are left seven points from safety with only two games remaining having now won just one of their last 19 Premier League games.
Chelsea, meanwhile, have now won 17 of their last 18 matches at Stamford Bridge and become only the third team in Premier League history to have won 300 games in front of their own fans.
More importantly, though, the seven-point gap to Tottenham means that Chelsea only need one win from their final three games of the season to clinch the title - something they could achieve before Spurs kick another ball.