Manchester United put their injury concerns to one side to keep their top-four dreams alive, while Liverpool suffered another slip-up on home soil in week 34 of the Premier League season.
The Red Devils are now just one point adrift of the Champions League places and only three behind their arch-rivals from Anfield with two games in hand.
Elsewhere, Hull City and Swansea City both recorded vital wins in their quest for survival.
The two clubs remain separated by just two points with four games each to play, with the Tigers currently having the edge and sitting in 17th place, just outside of the relegation zone.
Here is your complete round-up of the goings on in England's top tier this weekend.
PREMIER LEAGUE SCORES
|Swansea City||2-0||Stoke City|
Away goals were in short supply on Saturday, with all four home sides keeping clean sheets. Bournemouth made quick work of Middlesbrough, who are now nine points adrift of safety with a game in hand, thanks to goals from Josh King, Benik Afobe, Marc Pugh and Charlie Daniels.
Fernando Llorente and Tom Carroll netted for Swansea and Lazar Markovic and Sam Clucas scored for Hull to compound Middlesbrough's woes, with Marco Silva's men now on a four-match winning run at the KCOM Stadium.
West Ham and Everton, two sides mediocre at home and away respectively, played out a predictable 0-0 draw at the London Stadium.
On Sunday, the tide turned in favour of the away teams, with Manchester United and Crystal Palace prevailing on the road.
Jose Mourinho's Red Devils saw Anthony Martial and Wayne Rooney edged them past Burnley, while Christian Benteke returned to haunt his former employers at Liverpool with a match-winning brace on Merseyside.
PREMIER LEAGUE TABLE
Middlesbrough look as good as doomed after falling further behind Hull and Swansea, who will both be simultaneously pleased with their victories and frustrated they have not gained or made up any ground on each other.
Everton sneak above Arsenal into sixth place after picking up a point, but that lead is unlikely to last given the Gunners have three games in hand. Bournemouth, meanwhile, are making a late bid for a top-half finish.
Manchester United are still very much in the mix for a top-four finish, while Liverpool will now be looking anxiously over their shoulder as the chasing pack bunches up behind them.
WHO NEEDS ZLATAN?
After seeing Zlatan Ibrahimovic pick up an injury which is threatening to end his season and possibly his time at Old Trafford, Manchester United needed others to step up to the mark.
Against Chelsea, Marcus Rashford was deployed in a central striking role and repaid that show of faith with an early goal in an energetic display.
Jose Mourinho shuffled his pack again away at Burnley on Sunday, and got the big calls right once more as Anthony Martial led a swift counter to open the scoring at Turf Moor - leaving Joey Barton trailing in his wake - while returning captain Wayne Rooney wrapped up a 2-0 win as he prodded a second over the line.
United are now just one point behind their arch-rivals Manchester City, who sit fourth, with a derby date at the Etihad Stadium on Thursday taking on added importance for two sides who enjoyed contrasting fortunes in their most recent outing - with City tasting FA Cup semi-final heartache against Arsenal.
BORO SET FOR SWIFT RETURN
After an 11-year stay in the Premier League, Middlesbrough spent seven seasons in the Championship before finally winning promotion back to the big time. The opportunity to re-establish themselves in the top flight, though, has been passed up.
Boro look almost certain to be back in the second division next season after their 4-0 humbling at the hands of Bournemouth, a talented back line featuring Calum Chambers and Ben Gibson no longer able to hold the fort while the attacking players in front of them misfire.
Gaston Ramirez was worse than simply profligate at the Vitality Stadium, receiving a second yellow card after just 20 minutes to effectively end the match. Boro were already 2-0 down at that point after goals from Josh King and Benik Afobe.
Given that Cristhian Stuani was the Teessiders' top scorer in the Championship last season with 11, it was not difficult to see goals being their downfall despite the arrival of Alvaro Negredo. Flanked by Ramirez and Stewart Downing on Saturday, the 31-year-old was never likely to solve the problem of Boro being completely devoid of pace up front.
ALSO OF NOTE...
After six games without a win, Swansea halted their descent by beating Watford but remain in trouble due to Hull's excellent form at home.
Marco Silva's men have two home games and two away games remaining, hosting Sunderland and Tottenham and travelling to Southampton and Crystal Palace. If they can beat Sunderland and find another win from one of their other three games, Swansea will need two wins and two draws from their last four just to draw level with them and put their fate down to goal difference.
Swansea play Manchester United and Sunderland away and Everton and West Brom at home. There are points to be had there, but having rode their luck just to beat Stoke - Marko Arnautovic missed a penalty shortly before Carroll's deflected effort doubled their lead - whether or not they are up to the task remains to be seen.