wrap up the win in the second half.
The league leaders, who are seven points clear of Tottenham with three games to play, can clinch the title with victory at West Brom on Friday Night Football.
Boro barely threatened and the defeat makes it a swift return to the Championship one year and one day since they earned promotion against Brighton.
Spurs' defeat at West Ham last Friday eased the pressure on Antonio Conte's side and they took full advantage to move within touching distance of a second title in three years.
They now need just three points from their remaining three matches, with the visit to The Hawthorns followed by home games against Watford and Sunderland.
Football Writers' Player of the Year N'Golo Kante missed out with a muscle injury, handing Cesc Fabregas the opportunity to start.
The Spaniard was at the centre of Chelsea's commanding first-half display, creating their first chance inside 120 seconds.
His fine pass picked out Alonso but the wing-back saw his left-foot volley hit the crossbar.
The near miss only delayed the inevitable as Fabregas dragged an effort wide and Alonso went close again before Costa opened the scoring for the seventh time this season, more than any other player.
The striker collected Fabregas' pass from the right and tucked home under Brad Guzan.
Middlesbrough continued to struggle with Chelsea's cross-field passes and Alonso profited to force a second 11 minutes before half-time.
The wing-back drilled a low strike across Guzan and the hapless Boro 'keeper diverted the ball into his own net.
The half-time interval did little to halt Chelsea's charge as Pedro rattled the bar and Costa almost turned in his second at the far post.
They grabbed their third through Matic on 65 minutes with Fabregas once again pulling the strings.
His flicked pass found the Serbian in space and Matic controlled on his chest before firing through Guzan's legs.
The assist makes Fabregas the first player in Premier League history to record at least 10 in six different seasons.
A miserable evening for Boro threatened to get worse as Chelsea pushed to add further punishment but the last big cheer of the evening was reserved for John Terry.
The club captain was introduced for David Luiz with six minutes remaining, capping a straightforward night for Conte's side.
Chelsea: Courtois (7), Azpilicueta (8), Luiz (8), Cahill (8), Moses (8), Fabregas (9), Matic (8), Alonso (8), Pedro (8), Costa (8), Hazard (8)
Subs used: Terry (6), Chalobah (6), Willian (6)
Middlesbrough: Guzan (4), Fabio (5), Chambers (4), Gibson (5), Friend (5), De Roon (5), Clayton (5), Forshaw (6), Traore (4), Negredo (5), Downing (5)
Subs used: Leadbitter (5), Bamford (5), Gestede (5)
Man of the match: Cesc Fabregas