Anthony Martial scored the only goal at Turf Moor to extend United’s unbeaten run to eight matches.
There were no shots on target in the first half, with Ashley Young, back from suspension, curling the best chance wide.
Nine minutes after the break Romelu Lukaku passed for Martial to smash in off the crossbar and score in three consecutive PL appearances for the first time.
Almost immediately Burnley came close to an equalizer but Johann Berg Gudmundsson’s free-kick hit the woodwork and bounced over.
The Clarets kept pushing in Sean Dyche’s 100th PL match as a manager but, despite a scare from James Tarkowski, United held on for the win.
A flying start and Eden Hazard’s double helped Chelsea hold off a battling Brighton & Hove Albion side with a 4-0 victory at Amex Stadium.
Two goals in the opening six minutes set up the win, with Hazard converting a deflected Victor Moses cross before Willian scored from Michy Batshuayi’s flick in a brilliant team move.
Chelsea goalkeeper Willy Caballero, in for the injured Thibaut Courtois, withstood heavy pressure from Brighton, saving superbly from Tomer Hemed before seeing Davy Propper’s header hit the post.
The visitors capitalized with Hazard’s solo effort and a late fourth for Moses, following substitute Charly Musonda’s lofted pass.
A lightning start set Arsenal on the way to a 4-1 win over in-form Crystal Palace at Emirates Stadium.
Arsenal scored three goals in the opening 13 minutes, the fastest a team have taken a 3-0 lead in a Premier League match since the Gunners did so themselves seven years ago.
Nacho Monreal netted the first from Granit Xhaka’s corner before the defender produced two assists for Alex Iwobi and Laurent Koscielny.
Alexandre Lacazette made it 4-0 on 22 minutes, latching on to Mesut Ozil’s delightful flick for Arsenal’s 500th PL goal scored by a Frenchman.
Monreal went off injured before half-time and, after the break, Petr Cech was denied a 200th PL clean sheet when Luka Milivojevic netted a consolation goal.
Jamie Vardy and Riyad Mahrez struck as Leicester City moved up to seventh place with a 2-0 victory over Watford at King Power Stadium.
Both teams had a number of chances in the opening stages as Vardy and Watford’s Abdoulaye Doucoure had efforts well saved.
Vardy was fouled in the penalty area by Molla Wague and picked himself up to convert the 39th-minute spot-kick.
Wilfred Ndidi had two efforts blocked in the second half, but Mahrez secured the three points in stoppage time, having been found by Adrien Silva.
It was Mahrez’s fifth goal in as many successive PL matches against the Hornets.
Theo Walcott marked his debut with an assist as Everton drew 1-1 with West Bromwich Albion at Goodison Park.
Jay Rodriguez gave West Brom the lead on seven minutes, latching on to Grzegorz Krychowiak’s through-ball to find the bottom corner.
There was a lengthy delay in the second half due to a suspected double leg fracture suffered by James McCarthy, who left the field on a stretcher.
Substitute Oumar Niasse pounced on Walcott’s cushioned header to level a minute after being introduced.
Salomon Rondon almost produced a winner from distance late on but his drive came back off the crossbar.
Second-half goals from Joe Allen and Mame Biram Diouf gave Paul Lambert a 2-0 victory over Huddersfield Town in his first match as Stoke City manager.
After a goalless opening 45 minutes, Charlie Adam’s floated pass found Eric Maxim Choupo-Moting, who cut back for Allen to finish.
Stoke doubled their lead on 69 minutes, when a sublime flick from Xherdan Shaqiri fell into the path of Diouf, who slotted beyond Jonas Lossl.
A first win in five matches moves Stoke out of the bottom three, in 17th place on 23 points. Huddersfield are one point and three places further ahead.
Substitute Chicharito rescued a point for West Ham United as they shared a 1-1 draw with AFC Bournemouth at London Stadium.
The Mexican striker, who replaced the injured Manuel Lanzini, equalized for the Hammers just 64 seconds after they had fallen behind to Ryan Fraser’s strike.
After a goalless first half, the visitors went ahead when Fraser skipped past Pablo Zabaleta and finished well past Adrian on 71 minutes.
The lively Marko Arnautovic, who had twice come close to scoring, had another shot blocked but this time Chicharito was well placed to follow up.
In Saturday’s late game, Sergio Aguero marked his 200th Premier League appearance with a hat-trick as Manchester City recorded a 3-1 victory over Newcastle United.
City took the lead on 34 minutes when Aguero flicked on Kevin De Bruyne’s in-swinging cross.
Karl Darlow saved superbly from De Bruyne before the interval and touched Raheem Sterling’s shot on to the post.
Aguero struck again from the penalty spot after Sterling had been felled by Javier Manquillo.
Jacob Murphy reduced the arrears on the break with a dinked 67th-minute finish.
But Aguero registered his 138th PL goal and his seventh hat-trick in the competition late on after a bamboozling run by Leroy Sane.
City restored their 12-point lead at the top. The Magpies lie 15th.