Southampton slipped deeper into relegation trouble as Crystal Palace and Huddersfield won to move away from the bottom three in the Premier League.
Olivier Giroud scored twice as Chelsea came from 2-0 down to win 3-2 and leave Saints five points from safety.
Tom Ince scored a 91st-minute winner to help Huddersfield beat Watford 1-0.
Wilfried Zaha netted twice as Palace beat Brighton 3-2, with all five goals scored in the opening 34 minutes, while Swansea drew 1-1 at home to Everton.
Southampton, who started the day three points from safety, were on track to give Mark Hughes his first league win since being appointed.
But substitute Giroud's introduction sparked Chelsea into life at St Mary's and the reigning Premier League champions moved within to seven points of fourth-placed Tottenham.
Huddersfield jumped to 14th, seven points clear of 18th-placed Southampton, while Palace are six points from danger in 16th.
Jordan Ayew's second-half goal earned Swansea a point against Everton at the Liberty Stadium, but they dropped to 17th, five points above Saints.
First-half strikes from Chris Wood and Kevin Long gave Burnley a 2-1 win over Leicester at Turf Moor to strengthen their grip on seventh place and their hopes of qualifying for the Europa League.
Seventh will be enough to qualify for Europe unless Southampton win the FA Cup.
The Clarets are only two points behind sixth-placed Arsenal, who face Newcastle at 13:30 BST on Sunday.
In Saturday's late kick-offs, third-placed Liverpool host Bournemouth at 17:30), while Manchester City can move closer to winning the Premier League as they travel to Tottenham at 19:45.
In the Championship, Cardiff beat Norwich 2-0 to move a point behind second-placed Fulham in the race for automatic promotion, with a game in hand. Fulham host Brentford at 17:30.
Sunderland missed the chance to register a much-needed win in their quest to avoid relegation as they were held to a draw by fellow strugglers Reading.
Source: BBC NEWS