A win at West Bromwich Albion on Friday will secure the title in coach Antonio Conte's first season at Chelsea.
Costa got his 20th league goal this season in the 23rd minute, controlling a ball lofted into the area by compatriot Cesc Fabregas and neatly nutmegging keeper Brad Guzan.
Two Spaniards were involved in Chelsea's second goal in the 34th minute, Cesar Azpilicueta lifting the ball to Alonso on the left, who squeezed it under Guzan, with the ball hitting the stranded keeper’s legs on the way into the net.
Nemanja Matic got the third goal in the 65th from another neat Fabregas chip to pile the misery on Boro, who will join north-east neighbours Sunderland in the second tier next season.
"This game was very important for us after Tottenham's defeat (at West Ham on Friday) and the possibility to take three points and go seven points clear of Tottenham," said Conte.
"This is a big step because there are three games and we need to take three points to win the title. We need another win and then to celebrate another fantastic target for us."
Chelsea have 84 points from 35 games followed by Tottenham Hotspur on 77 points also with three games remaining.
Chelsea were without their injured France international N'Golo Kante, who won this season's PFA and football writers' Player of the Year awards, but Spaniard Cesc Fabregas controlled the midfield with almost languid confidence in his absence.
By the second half Conte's team were buzzing and an increasingly beleaguered Middlesbrough side rarely managed to move out of their own half as they headed for the trap door.
Despite his side's total control of the game, the ever feverish Conte prowled the touchline as usual, gesticulating wildly and yelling commands to his players.
Chants of "Antonio, Antonio" reverberated around the ground as the whistle went and the Italian hugged his assistants before applauding and shaking hands with the despondent Boro players.
(Reporting by Clare Lovell; Editing by Ken Ferris)
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