Match ends, Swansea City 0, Chelsea 1.
Swansea City 0-1 Chelsea
Chelsea "kept their Champions League hopes alive" with a win at Swansea that left them two points behind fourth-placed Tottenham, said manager Antonio Conte.
Cesc Fabregas' 50th Premier League goal was enough to secure victory for the Blues, who have three matches remaining.
Tottenham play their game in hand against Watford on Monday.
"We got three points and it was very important for us if we wanted to achieve Champions League football," Conte said.
"The situation is not in our hands, but the only way to put a bit of pressure on Tottenham is to get three points in every game.
"It won't be easy, we know they will play against Watford. But you know very well football is not simple.
"It is not simple to win a game in this league but for sure our situation is not in our hands."
Swansea are now only a point ahead of Southampton in the relegation zone after the Saints beat Bournemouth earlier in the day.
The result earned a reprieve for West Brom, who would have been relegated to the Championship had Swansea drawn.
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- What happened in the Premier League on Saturday
- Moore's magic touch - Saturday's best stats
Hazard leading charge for top four
Conte will have greater confidence about returning to the Champions League if Eden Hazard can continue his late-season surge in form.
The Belgium international was Chelsea's standout performer in last weekend's FA Cup semi-final win over Southampton, and was every bit as instrumental here.
Fabregas' curling finish was a fitting way to bring up his half-century of goals in the Premier League, but it was Hazard's clinical exploitation of a Swansea error that made it, pouncing on Andy King's sloppy loss of possession, weaving his way forward and picking out the right pass.
Antonio Rudiger may well have given Hazard his second assist soon after, only for the defender to spurn the chance at the far post following a corner.
Hazard was at the heart of Chelsea's best threatening moves and made the most of some naive defending as Swansea afforded him plenty of space for surging runs on the counter.
With an almost flawless finish to their league campaign needed to overtake either Tottenham or Liverpool, Chelsea will need the former Footballer of the Year to remain as dependable and decisive as he was here.
Swans still have fate in their own hands
It appears destined the Liberty Stadium will decide this year's survival battle.
Following their trip to Bournemouth next weekend, Swansea's final two fixtures are in South Wales - both against direct rivals.
Southampton could yet have the chance to overtake Swansea when they visit the Liberty on Tuesday, 8 May before Stoke travel across the border on the final day of the season.
Swans boss Carlos Carvalhal is clearly aware of the importance of maintaining belief at home having taken the scalps of Liverpool and Arsenal since his appointment, as well as registering wins over Burnley and West Ham.
Both the Portuguese and chairman Huw Jenkins issued rallying cries to supporters before the game, but while the fans stuck behind their side, it took too long for Swansea to add a real sense of attacking urgency.
Chelsea - who had skimmed the top of the bar after Alfie Mawson cleared a Olivier Giroud header - did not see goalkeeper Thibaut Courtois really tested until the final 15 minutes, when Andre Ayew fizzed in a shot. Kyle Naughton, Tom Carroll and Wayne Routledge also went close.
Swansea claimed for a penalty when Gary Cahill challenged Nathan Dyer, with the Chelsea defender also involved in a clash with Jordan Ayew in injury time after a late barge from the Ghanaian.
With Southampton seemingly having the momentum, Swansea might have to take more chances at Bournemouth next weekend to avoid a do-or-die final week.
"When we arrived we were five points away from the teams above," Carvalhal said.
"We depended on others to try and get out of this position. In this moment we just depend on us, it's something we have in our hands.
"We must look to the games that we have - Bournemouth, Southampton and Stoke - and try to achieve the necessary points to stay in the division. We will do our job."
Man of the match - Eden Hazard (Chelsea)
Fabregas joins the elite
- Antonio Conte has won 50 of his 73 Premier League games in charge - only Jose Mourinho (63) and Pep Guardiola (69) reached 50 wins as manager in the competition in fewer games.
- Swansea have failed to score in 16 different Premier League games this season; in no campaign have they failed to score in more in the competition (also 16 in 2012-13).
- Chelsea have won three consecutive away league games for the first time since September (a run of five).
- Meanwhile, Swansea have lost consecutive league games for the first time under Carlos Carvalhal, last doing so in Paul Clement's final two games in charge in December.
- Cesc Fabregas became only the fourth player to score 50+ goals and register 100+ assists in the Premier League, and first non-British player to do so (after Ryan Giggs, Wayne Rooney and Frank Lampard).
- Olivier Giroud hasn't scored in any of his last six Premier League starts, with his last six goals in the competition all coming as a sub
Swansea travel to Bournemouth next Saturday while Chelsea take on Liverpool in a key match in the Champions League push at Stamford Bridge next Sunday.
- 5van der Hoorn
- 52RobertsSubstituted forCarrollat 63'minutes
- 4Ki Sung-yueng
- 24KingSubstituted forDyerat 58'minutes
- 16OlssonSubstituted forRoutledgeat 81'minutes
- 19A AyewBooked at 15mins
- 18J Ayew
- 35Renato Sanches
- 15MosesBooked at 30mins
- 4FàbregasSubstituted forPedroat 81'minutes
- 10E HazardSubstituted forWillianat 81'minutes
- 18GiroudSubstituted forMorataat 85'minutes
- Jonathan Moss
- Shots on Target
Second Half ends, Swansea City 0, Chelsea 1.
N'Golo Kanté (Chelsea) wins a free kick in the defensive half.
Foul by Jordan Ayew (Swansea City).
Álvaro Morata (Chelsea) wins a free kick in the attacking half.
Foul by Mike van der Hoorn (Swansea City).
Attempt saved. Wayne Routledge (Swansea City) right footed shot from the centre of the box is saved in the bottom left corner. Assisted by Jordan Ayew.
Attempt missed. Tom Carroll (Swansea City) left footed shot from outside the box is close, but misses to the right. Assisted by Sam Clucas.
Offside, Chelsea. Willian tries a through ball, but Álvaro Morata is caught offside.
Attempt saved. Alfie Mawson (Swansea City) right footed shot from outside the box is saved in the bottom left corner. Assisted by Sam Clucas.
Substitution, Chelsea. Álvaro Morata replaces Olivier Giroud.
Tiemoué Bakayoko (Chelsea) wins a free kick in the defensive half.
Foul by Kyle Naughton (Swansea City).
Substitution, Chelsea. Pedro replaces Cesc Fàbregas.
Substitution, Swansea City. Wayne Routledge replaces Martin Olsson.
Substitution, Chelsea. Willian replaces Eden Hazard.
Attempt missed. Eden Hazard (Chelsea) left footed shot from outside the box misses to the right. Assisted by Olivier Giroud with a headed pass.
Attempt saved. Kyle Naughton (Swansea City) right footed shot from outside the box is saved in the bottom right corner.
Attempt missed. André Ayew (Swansea City) left footed shot from the centre of the box misses to the left. Assisted by Tom Carroll.
Attempt missed. Martin Olsson (Swansea City) left footed shot from the left side of the box misses to the right. Assisted by Tom Carroll with a cross.
Attempt missed. Victor Moses (Chelsea) right footed shot from very close range is close, but misses to the right.
Attempt blocked. Cesc Fàbregas (Chelsea) right footed shot from the right side of the box is blocked.
Eden Hazard (Chelsea) wins a free kick in the defensive half.
Foul by Alfie Mawson (Swansea City).
Attempt missed. Jordan Ayew (Swansea City) header from very close range is too high following a corner.
Corner, Swansea City. Conceded by César Azpilicueta.
Attempt saved. Emerson (Chelsea) right footed shot from the centre of the box is saved in the bottom right corner. Assisted by Olivier Giroud.
Attempt saved. Eden Hazard (Chelsea) header from the centre of the box is saved in the bottom right corner. Assisted by César Azpilicueta with a through ball.
Attempt missed. Tiemoué Bakayoko (Chelsea) right footed shot from outside the box misses to the left. Assisted by César Azpilicueta.
Foul by N'Golo Kanté (Chelsea).
André Ayew (Swansea City) wins a free kick on the left wing.
Substitution, Swansea City. Tom Carroll replaces Connor Roberts.
Offside, Chelsea. Victor Moses tries a through ball, but Cesc Fàbregas is caught offside.
Substitution, Swansea City. Nathan Dyer replaces Andy King.
Foul by Tiemoué Bakayoko (Chelsea).
Ki Sung-yueng (Swansea City) wins a free kick in the defensive half.
Attempt missed. André Ayew (Swansea City) header from very close range is too high. Assisted by Martin Olsson with a cross.
Olivier Giroud (Chelsea) wins a free kick in the defensive half.
Foul by Alfie Mawson (Swansea City).
Foul by Tiemoué Bakayoko (Chelsea).