United are gearing up for next week’s Europa League final against Tottenham, but ran the Blues close as Ruben Amorim named a stronger line-up than expected.
However, the Red Devils ultimately slipped to an 18th league defeat of the season and remain 16th in the table.
Chelsea and Aston Villa, who beat Spurs 2-0 in Friday’s other Premier League match, both leapfrog Manchester City into the top five.
City have a game in hand at home to Bournemouth on Tuesday as they face Crystal Palace in Saturday’s FA Cup final.
But Chelsea will be assured of a return to Europe’s top competition after a two-year absence should they beat Nottingham Forest on the final day of the season.
United’s miserable season can only be saved by victory in Bilbao in five days’ time but Amorim resisted the temptation to rest his key men at Stamford Bridge.
Captain Bruno Fernandes, Amad Diallo and Casemiro were among those to start as Amorim named close to his strongest available side.
Chelsea were without a recognised striker due to Nicolas Jackson’s suspension and the continued absence due to injury of Christopher Nkunku.
Teenager Tyrique George, a natural winger, was forced to lead the line in an unfamiliar role up front and the Blues’ lack of a focal point saw them easily snuffed out.
United’s bright start was nearly rewarded when Harry Maguire fired in Fernandes’ inviting cross, only for a VAR review to show the centre-back had strayed centimetres offside.
Chelsea’s best effort before half-time saw Reece James’ long-range curler come back off the post.
Cole Palmer was the one creative spark for Enzo Maresca’s men and nearly produced a moment of magic with a dipping shot that Andre Onana palmed into the path of Enzo Fernandez.
The Argentine somehow fired over with the goal gaping and was grateful to the offside flag for sparing his blushes.
A match littered with stoppages for minor injuries and VAR checks did not allow either side to hit their rhythm.
United started strongly again in the second period but failed to make it count as Mason Mount fired wide on his return to Stamford Bridge, before Fernandes blasted over.
On the hour mark a VAR review came to the visitors’ aid when referee Chris Kavanagh initially pointed to the penalty spot after a challenge between George and Onana.
However, replays showed the goalkeeper clearly played the ball and the spot-kick was overturned.
With Chelsea’s forward players lacking inspiration, the match-winning moment was created and converted by their full-backs 20 minutes from time.
James turned nearly on the edge of the box and clipped in a cross that Cucurella met perfectly to head in his seventh goal of the season.
Noni Madueke should have doubled the lead within seconds when he skewed wide after a slick pass from Palmer.
That miss could have been costly had Robert Sanchez not denied Diallo at his near post.
But Chelsea held their nerve to edge towards the brink of a return to the riches and glamour of the Champions League.
In the early game, Aston Villa bolstered their bid to qualify for the Champions League as they beat Tottenham 2-0 to climb into fifth place in the Premier League.
Unai Emery’s side had no margin for error at Villa Park as the race for top five places approaches a dramatic conclusion.
Second-half goals from Ezri Konsa and Boubacar Kamara ensured Villa’s battle to play in Europe’s elite club competition will go down to the last day of the season. (AFP)
