Spaniard Juan Mata's late strike secured Chelsea a 1-0 victory at Manchester United on Sunday to leave the Londoners poised for a top-four finish and a Champions League place next season.
Mata settled a poor game when his left-foot shot in the 87th minute took a slight deflection off United defender Phil Jones and beat keeper Anders Lindegaard.
United defender Rafael was sent off in the 89th minute for kicking out at David Luiz.
Chelsea moved up to third place on 68 points from 35 games, one ahead of Arsenal, who have played a game more, and three ahead of Tottenham Hotspur, who have also played 35.
In the earlier game, Everton's slim hopes of playing European football next season receded further after they drew 0-0 at city rivals Liverpool in a derby short on entertainment.
Everton, who have drawn 15 Premier League games this season - more than any other side - remain sixth on 60 points.
They have just two games remaining and have slim hopes of securing the Europa League place for the side finishing fifth. Liverpool are seventh on 55 points.