Cesc Fabregas boosted Chelsea's bid to qualify for the Champions League as his superb strike clinched a 1-0 win at Swansea, while West Brom kept alive their slender hopes of avoiding Premier League relegation with a 1-0 victory against Newcastle on Saturday.
Spain midfielder Fabregas gave Chelsea the perfect start at the Liberty Stadium when he took an Eden Hazard pass and fired home for his 50th Premier League goal in the fourth minute.
That was enough to secure a third successive league win for Antonio Conte's fifth-place side, who move within two points of Tottenham ahead of their fourth-placed London rivals' clash with Watford on Monday.
It was a damaging defeat for Swansea and Carlos Carvalhal's team are now only one point above the relegation zone with three games left.
West Brom would have been relegated to the Championship if they had failed to take maximum points at St James' Park, but Matt Phillips bagged the first-half winner to keep them in with a shout for a few more hours at least.
Jake Livermore's pin-point pass sent Phillips racing clear of the Newcastle defence and the winger lashed past Martin Dubravka in the 29th minute.
Bottom club Albion are five points from safety and an eight-season stay in the top flight would still be over if they fail to take maximum points from their last two games against Tottenham and Crystal Palace.
Interim boss Darren Moore said: "We can't control other results. There's nothing we can do.
"We're just doing everything we can at our end. The boys have done really well since I came in."
Dusan Tadic breathed new life into Southampton's bid to beat the drop as his double sealed a crucial 2-1 success in the south-coast derby against Bournemouth at St Mary's.
Mark Hughes' third-bottom side took the lead when Serbia winger Tadic finished off a rapid counter-attack from Mario Lemina in the 25th minute.
Joshua King equalised for Bournemouth with a close-range finish after Saints failed to clear a corner in first-half stoppage-time.
But Southampton restored their advantage in the 54th minute thanks to a gem from Tadic, who embarked on a fine solo run before finishing in clinical fashion.
It was Southampton's first league win in five matches since Hughes replaced the sacked Mauricio Pellegrino last month.
Southampton, hoping to extend their six-season stint in the top flight, are one point adrift of safety with three games left.
"This is one of the most important wins in our history," Tadic said. "We are fighting for our lives and we showed that. We knew we had to win, no matter how we played."
Jurgen Klopp admitted Liverpool were below their best as they prepared for the Champions League semi-final, second-leg clash with Roma by settling for a goalless stalemate against Stoke at Anfield.
The Reds head to Rome on Wednesday with a 5-2 advantage from the first leg and do so on the back of a tame encounter they controlled for long periods without over-exerting themselves in search of a winner.
The draw left Liverpool needing a maximum of four points from their remaining two matches to guarantee qualification for the Champions League via a top-four finish.
"I saw everything I wanted to see except fluency and fun and joy," Klopp said. "It is hard on a day like this. But no-one was seriously injured so we carry on."
Second-bottom Stoke are three points from safety, but boss Paul Lambert said: "I could not have asked for any more. If the lads had played like that from the start of the season, I would not be here and they would be cruising in the league."
Crystal Palace are on course to avoid relegation after thrashing Leicester 5-0 at Selhurst Park.
Everton pushed Huddersfield deeper into the relegation mire as goals from Cenk Tosun and Idrissa Gueye earned a 2-0 win at the John Smith's Stadium.Burnley, in seventh and chasing a Europa League place, were held to a 0-0 draw by Brighton at Turf Moor.