Antonio Conte’s side had been knocked out of first place by Arsenal on Saturday, but they reopened a three point lead over the Gunners thanks to Costa’s superb finish in the closing stages at Stamford Bridge.
Costa’s 12th league goal in his last 15 appearances gave Chelsea a ninth successive league win for the first time since 2007.
The west Londoners have scored 23 goals and conceded just two in that blistering streak, raising hopes of a Premier League title in Conte’s first season in charge.
Henrikh Mkhitaryan scored his first Premier League goal for Manchester United but was later stretchered off as Jose Mourinho’s side claimed a timely 1-0 victory over Tottenham Hotspur on Sunday.
Mkhitaryan converted Ander Herrera’s through ball just before the half hour to end a frustrating run of four successive home draws and give United just a second win in nine league matches.
It was his second goal in two games, after he opened his account for the club against Zorya Luhansk on Thursday, but he exited on a stretcher after turning his ankle in a late challenge by Danny Rose.
United remain 13 points below leaders Chelsea, but closed to within three points of fifth-place Spurs and six off Manchester City, who occupy the fourth and final Champions League berth.
Elsewhere, Sofiane Boufal’s first Premier League goal was one to remember as his stunning 25-yard strike sealed Southampton a 1-0 win over Middlesbrough.
Boufal joined Saints in a club-record deal worth £21million last summer and the Moroccan’s moment of magic proved decisive at St Mary’s in a game of few clear-cut chances.
Southampton’s victory moves them up to 10th in the table while Middlesbrough remain 16th, after just one win in five games.
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