Manchester United claimed their 600th Premier League victory with a 3-1 win against managerless Middlesbrough at the Riverside Stadium on Sunday.

Marouane Fellaini's first league goal of the season, a thundering strike from Jesse Lingard and an injury-time Antonio Valencia finish secured maximum points for Jose Mourinho's side and lifted into fifth place in the table, piling the misery on Boro at the end of a week in which Aitor Karanka lost his job.

Mourinho – whose men are now unbeaten in 18 top-flight matches – had claimed that a draw would be a good result for his side, given their exhaustive exploits in the FA Cup and Europa League this week, and Boro certainly started in the ascendancy against a United side shorn of a number of first-team regulars.

But Fellaini's 30th-minute header gave the visitors a stranglehold that they never looked likely to relinquish and Lingard's emphatic strike in the second half left the home side on course for a 14th defeat in 28 league games this season.

Rudy Gestede got a goal back in the closing stages – Boro's first in 509 minutes of league action – but Valencia pounced on a horror moment for ex-United goalkeeper Victor Valdes to ensure United move to within three points of the Champions League places, having played a game fewer than fourth-placed Liverpool, who face Manchester City later on Sunday.

The pressure is growing on second-bottom Boro, however, with five points now separating them from safety, though they do have a game in hand on Hull City and Swansea City above them.

Valdes' outing started well as he made a fine low stop to deny Marcus Rashford six minutes into the contest after the young striker raced in behind the defence onto Lingard's pass.

Boro – guided by stand-in boss Steve Agnew – enjoyed by far the better of the opening exchanges, though, and David de Gea produced a smart save to his left to prevent Gaston Ramirez's fierce strike from breaking the deadlock.

Valdes proved the home side's saviour again in the 22nd minute, sticking out a leg to deny Rashford from point-blank range after a superb United move before reacting well to keep out Valencia's rebound.

United were beginning to cause real problems for the home defence and Fellaini took full advantage when the next chance came his way. Ashley Young curled a deep cross towards the far post from the right and the Belgium international escaped Fabio before nodding the ball high into Valdes' net.

Boro began the second half in much the same fashion as they did the first, enjoying close to 57 per cent of the possession, but without finding a way to cause any trouble for De Gea, with United's defence staying resolute.

Mourinho's side continued to look a threat on the break, with Rashford and Valencia twice causing panic in the Boro box with runs down the right, and it was from another counter-attack that Lingard doubled the lead just past the hour mark.

Collecting Juan Mata's pass, Lingard raced unchallenged through the middle of the home side's half before blasting a shot from 20 yards that swerved beyond the static Valdes and into the top-right corner.

Agnew threw on Gestede to give Alvaro Negredo some support and the striker gave his side a lifeline when he capitalised on Chris Smalling's error to slot past De Gea from close range.

Boro laid siege to United's goal late on, but Valencia made the points safe in the dying seconds when Valdes slipped as he prepared to clear a backpass, allowing the United full-back the simple task of finishing into an empty net – much to the delight of Mourinho as the Portuguese made his way down the tunnel.