Jesse Lingard scored twice but Paul Pogba was sent off as Manchester United beat Arsenal 3-1 in a breathless match at the Emirates Stadium.
United were ahead after only four minutes when Laurent Koscienly's crossfield pass was cut out by Antonio Valencia, who played a one-two with Pogba and beat Petr Cech, adding another after 11 when Anthony Martial's flick freed Lingard.
Alexandre Lacazette was denied by De Gea, who deflected his strike onto the bar and Granit Xhaka sidefooted fractionally wide. De Gea then tipped away after the ball cannoned off Lukaku following an Arsenal free kick, but less than five minutes into the second period Lacazette lashed in to halve the deficit.
After 56 minutes Arsenal looked certain to level when Lacazette's low shot was superbly saved by De Gea and Sanchez, following up, was somehow denied by the keeper. But seven minutes later United were two clear again, Pogba setting up Lingard.
But their night was marred when Pogba was shown a straight red card for a stamp on Hector Bellerin, who had fouled Ashley Young moments earlier, ruling him out of the forthcoming Manchester derby.
Devastating counter-attacking held the key for Liverpool as they continued their good run with a thumping 5-1 victory at Brighton.
Liverpool dominated the early possession and took the lead on the half-hour when Emre Can headed home, doubling their advantage two minutes later when a flowing counter gave Firmino the chance to finish from close range.
The home side almost got back into it when Mignolet saved Murray's close-range effort early in the second half -- but from that stop, Liverpool produced another brilliant counter, with Firmino again rounding it off to make it 3-0.
Brighton got one back almost immediately when Jordan Henderson pushed Shane Duffy in the area and Murray converted from the penalty spot.
The home striker continued to cause problems for Liverpool and almost cut the deficit to 3-2 when he swivelled to send an effort a whisker past the post, but Coutinho fired home a free kick and then Lewis Dunk put through his own goal.
A sparkling two-goal performance from Eden Hazard helped Chelsea come from a goal down to beat Newcastle 3-1 at Stamford Bridge and go level on points with second-placed Manchester United.
Newcastle profited from defensive uncertainty after 12 minutes when a long ball over the top enabled Jacob Murphy to threaten and, when Thibaut Courtois thwarted him, Dwight Gayle slid the ball home.
That woke the champions up and, after Andreas Christensen had hit a post, they were level with 21 minutes gone when Eden Hazard sent his finish into the ground and over Newcastle keeper Karl Darlow.
Shortly after the half-hour, resurgent Chelsea were in front as Victor Moses fizzed in a cross and Alvaro Morata thumped the header past Darlow, and Hazard was on the mark again when he dinked home a penalty given for Matt Ritchie's 74th-minute foul on Victor Moses.
Ten-man Tottenham Hotspur took a point from Watford in a 1-1 draw at Vicarage Road.
Spurs could have taken an early lead when a teasing low cross from Kieran Trippier just evaded Harry Kane, but the hosts broke through after 13 minutes when Tom Cleverley's corner was headed home by an unmarked Christian Kabasele.
Richarlison appeared to be fouled by Ben Davies inside the area moments before Spurs raced to the other end to level after 25 minutes when Son Heung-Min turned in Christian Eriksen's cross.
The visitors were down to 10 men after 52 minutes when Davinson Sanchez was punished for catching Richarlison in the face with an arm.
Abdoulaye Doucoure was soon agonisingly close to re-establishing their lead when his cracking strike hit the inside the post, and Watford appealed in vain for a late penalty when they felt a Richarlison cross had been handled.
Sam Allardyce's reign at Everton got off to a perfect start as his new side saw off Huddersfield 2-0 at Goodison Park.
Little of note happened in the early stages at a subdued Goodison Park, with only an Aaron Lennon break for Everton and an Aaron Mooy cross for Huddersfield causing any problems.
Two minutes into the second half, Dominic Calvert-Lewin's flick enabled Gylfi Sigurdsson to run on and slot home the opening goal, but Huddersfield were close to responding quickly when Tom Ince hit the side netting.
And Everton sealed a second successive home win when Calvert-Lewin raced away and saw his shot deflected into the net.
Leicester held on for a 1-0 home win over Burnley as they continued their climb up the table.
The home side took only six minutes to break through against their high-flying opponents as a Riyad Mahrez cross diverted by keeper Nick Pope and Demarai Gray bundled home before colliding painfully with the post.
Burnley stormed back, Chris Wood and Ben Mee missing chances, but Leicester should have doubled their lead with the hour approaching when neither Jamie Vardy nor Wes Morgan could make a clean connection inside the six-yard box.
They came even closer when Gray hit the post but, despite late Burnley pressure, made it two wins in a week following victory over Spurs.
Alan Pardew had to settle for a point as his new West Bromwich Albion side were held to a 0-0 draw by his old one Crystal Palace at The Hawthorns.
Pardew's reign almost got off to a perfect start with eight minutes on the clock, but Hal Robson-Kanu could only hook over after a deflected Kieran Gibbs cross had found him.
Wilfried Zaha was causing problems for Palace as the Londoners began to get on top, with Christian Benteke then heading straight at the keeper. Jay Rodriguez came closest for the Baggies when he fired narrowly wide as the home side got into their stride after the break.
Stoke came back from a goal down to leave Swansea bottom of the table with a 2-1 home win.
The struggling Swans made a dream start, needing only three minutes to take the lead when Martin Olsson's cross from the left was superbly converted by ex-Stoke striker Wilfried Bony.
But with 36 minutes gone, the home side were back on level terms when Leroy Fer was dispossessed in a dangerous area and Joe Allen set up Xherdan Shaqiri -- and they led moments later when Mame Biram Diouf struck.
Swansea had chances to find an equaliser after the break, but Peter Crouch came close to adding a third as Stoke saw it through.