Fulham, Crystal Palace and West Brom also picked up wins on the road in their battle to beat the drop.
The Cottagers clambered off the bottom of the table as substitute Hugo Rodallega netted late on - his first league goal in 15 months - to seal a 2-1 win at Aston Villa, while Cardiff are now six points from safety after suffering a heavy 3-0 defeat to Palace.
Pepe Mel's Baggies claimed a 1-0 victory at Norwich to move above the Canaries on goal difference. Sunderland now prop up the top flight ahead of their trip to Tottenham on Monday night.
George Boyd's first-half header also secured Hull City a 1-0 home win over Swansea.
At the top of the table, Manchester City saw off Southampton 4-1 at the Etihad in the day's early kick-off but it was Chelsea who climbed back to the summit after a 3-0 teatime victory over Stoke City.
Chelsea started the day two points behind leaders Liverpool but bounced back from a couple of disappointing defeats to see off Stoke and stay very much in contention for the title.
Making his first Premier League start, Mohamed Salah fired Chelsea in front on 32 minutes with a powerful first-time strike after good work by another January signing, Nemanja Matic, down the left wing.
Salah made another important contribution just after the hour mark when he charged into the area and was brought down by Andy Wilkinson, giving Frank Lampard the chance to make it 2-0 from the penalty spot. Asmir Begovic saved Lampard's initial effort but the midfielder followed up to put the rebound away.
The impressive Willian then sealed victory for a dominant Chelsea when he curled a wonderful finish past Begovic in the 72nd minute.
Manchester City comfortably avoided another slip-up in their pursuit of leaders Liverpool as they beat Southampton 4-1 at the Etihad Stadium.
Manuel Pellegrini's men dropped two points at Arsenal last weekend, but got off to a perfect start in the day's early kick-off when Yaya Toure scored his 18th Premier League goal of the season after just three minutes, following Jose Fonte's foul on Edin Dzeko.
Saints - and England - suffered a major blow when Jay Rodriguez had to be stretchered off with what looked to be a serious knee injury and although Rickie Lambert levelled with a spot-kick, the hosts went in at the break two goals up. Mauricio Pochettino was still complaining about City's second - Samir Nasri finishing off a pass from David Silva, who was considerably offside when he received a Dzeko flick - when the Bosnian headed home Aleksandar Kolarov's left-wing delivery.
Stevan Jovetic made sure of the points when he came off the bench and converted a Jesus Navas cross in the 81st minute. City's next fixture is the trip to face Liverpool at Anfield on Sunday week, live on Sky Sports 1HD.
With Manchester United's Champions League quarter-final second leg away to Bayern Munich taking place on Wednesday night, David Moyes made eight changes to his starting line-up at Newcastle, with Juan Mata one of the players coming in.
The Spain star scored the only goal of the first half with a 39th-minute free-kick, awarded after Dan Gosling - making his first Premier League start since December 2011 - fouled Darren Fletcher on the edge of the area. Mata stepped up to score with a terrific curling effort into the top corner of Rob Elliot's net.
Then on 50 minutes, Mata was on the scoresheet again. A long ball from the back down the right channel was taken on into the box by Javier Hernandez, and then squared for the arriving Mata. Shinji Kagawa threatened to get in the Spaniard's way, but Mata kept his composure and dummied the keeper before coolly rolling the ball into the net.
Hernandez got in on the act on 64 minutes, sliding the ball underneath Elliot after Kagawa had played him in at the near post. Adnan Januzaj netted a fourth in the 90th minute after a lovely backheel one-two with Mata in the area.
Despite Manchester United's rise, Tottenham could regain sixth spot if they can beat Sunderland at White Hart Lane in Monday Night Football.
With Christian Benteke's season over due to injury and Gabby Agbonlahor ill, striker Grant Holt started up front for Aston Villa for only the second time since his January loan switch from Wigan. Mahamadou Diarra - back at Fulham for a second spell - was among three changes made by Felix Magath after the defeat by Everton.
Teenage striker Cauley Woodrow had the best opportunity of the first half when Joe Bennett's poor pass was picked up by the 19-year old. He nutmegged Nathan Baker, but could not beat Brad Guzan from the angle.
Magath introduced Rodallega and Ashkan Dejagah in place of Woodrow and Patjim Kasami at the start of the second half
Villa hit the post five minutes after the break when Leandro Bacuna thundered a shot towards goal, as Fulham goalkeeper David Stockdale got an important touch to divert the ball onto the woodwork.
Sascha Riether made a timely block to deny Andreas Weimann, and then Dejagah set up Fulham's opening goal just after the hour mark when he jinked infield from the right and passed across the top of the box for Kieran Richardson to lash a shot past Guzan.
When Marc Albrighton swung in a 70th-minute corner, Holt peeled off the Fulham defence and headed down between Weimann and Steve Sidwell, with Richardson unable to keep the ball out.
Soon afterwards, Lewis Holtby was denied by an incredible goalline clearance from Matt Lowton - the Goal Decision System replays suggesting there was only millimetres in it - and then Stockdale saved brilliantly from Weimann at the other end.
But when Holtby swung in a cross from the right four minutes from time, Rodallega connected with a glancing header to give Fulham their first away win since Boxing Day. They jump two places to 18th, but are still five points from safety.
Wilfried Zaha started for Cardiff against his former club, while ex-Bluebirds Joe Ledley and Cameron Jerome were part of a Crystal Palace side which unsurprisingly showed no changes following the 1-0 win over Chelsea last weekend.
Half an hour had elapsed before Jason Puncheon drew first blood with his fourth goal of the season. Palace attacked down the right and Ledley bided his time on the edge of the box before returning a pass to Puncheon, who had snuck into the area. The winger eluded a challenge from Andrew Taylor before burying the ball past David Marshall.
In the 71st minute, Marshall saved Marouane Chamakh's downward header from a Puncheon free-kick but he could only put the ball at the feet of Ledley, who scored from close range - but declined to celebrate against his hometown team.
Puncheon wrapped up the three points when he curled in a left-footed shot from 20 yards in the 88th minute. The win takes Palace up two places to 14th, seven points clear of danger, while Cardiff are six points adrift in 19th.
At Norwich, James Morrison was in Pepe Mel's West Brom starting line-up, while Saido Berahino - who was confronted by his Scottish team-mate in the dressing-room bust-up which followed Albion's 3-3 draw with Cardiff- was named on the bench.
Albion gained a 16th-minute advantage when Morgan Amalfitano - scorer of a stunning goal against Cardiff last weekend - found the target again. James Morrison and Youssouf Mulumbu were both involved in a neat passing move which ended with the Frenchman sliding in a firm low shot across John Ruddy from the edge of the box.
Ruddy saved well from Matej Vydra in the second half, and Ben Foster also made a smart stop to keep out Johan Elmander's header. Norwich came closest at Carrow Road, however, when Robert Snodgrass struck the crossbar with a 30-yard free-kick.
The Baggies rise above the Canaries into 16th spot. The two clubs are level on 32 points.
George Boyd was back after a three-match suspension for Hull City as they hosted Swansea in a mid-table encounter at the KC Stadium. Maynor Figueroa also came in, with back injury victim Tom Huddlestone held back just in case ahead of next weekend's FA Cup semi-final against Sheffield United at Wembley.
In a tight first half, it was Boyd who opened the scoring when he powered home a header at the back post from Liam Rosenior's right-wing cross after 39 minutes.
The best opportunity for Swansea to level fell to substitute Nathan Dyer midway through the second half, but he fluffed his lines by firing wide after being picked out by a cross fed to the edge of the box by fellow winger Wayne Routledge.
Hull move up to 12th, leapfrogging the Swans in the process.