Everton's 100 per cent record under Carlo Ancelotti came unstuck in a 2-1 defeat at Manchester City, but Mikel Arteta and David Moyes got their first victories in charge of West Ham and Arsenal, respectively.
Arteta, whose Arsenal side put in a much-improved performance, claimed a 2-0 victory over Manchester United, while West Ham crushed Bournemouth 4-0 on the Scot's return to the club.
Danny Ings scored the only goal of the game as Tottenham suffered a surprise 1-0 loss to Southampton, while top-four rivals Chelsea were held 1-1 by Brighton and Hove Albion.
There were also wins for Aston Villa, Leicester City and Watford, along with a draw between Norwich City and Crystal Palace, as 2020 got off to an eventful start, with VAR once again proving to be a big talking point.
Jesus at the double while Noble inspires Hammers
Ancelotti delivered back-to-back wins in his opening two games as Everton boss and his Toffees side frustrated City for 51 minutes in a tight game at the Etihad.
The deadlock was broken by Gabriel Jesus, who scored with a scintillating right-footed shot into the top corner from the edge of the box, and he connected with a Riyad Mahrez pass to score with a low shot seven minutes later.
Richarlison turned Theo Walcott's deflected cross into the net at the far post with 19 minutes left to give Everton hope, but they failed to find an equaliser and suffered a first defeat in six league games.
At London Stadium, Moyes produced a markedly improved performance from West Ham as they strolled to a 4-0 victory over Bournemouth, with captain Mark Noble scoring a brace.
Noble scored with a deflected shot with 17 minutes played and Sebastien Haller found the net via a bicycle kick before the Hammers' skipper converted a first-half penalty.
Felipe Anderson netted with a neat finish when Declan Rice found him with a measured pass, West Ham's fourth wrapping up their biggest margin of victory in the Premier League for over a decade.
Arteta off the mark at jubilant Emirates
Arsenal's home faithful were treated to a dominant display and a 2-0 victory over Manchester United that harked back to the days when Patrick Vieira and Dennis Bergkamp tormented the Red Devils in north London.
Nicolas Pepe opened the scoring in the eighth minute, firing home from 12 yards after Pierre-Emerick Aubameyang and Sead Kolasinac combined to great effect down the Arsenal left.
Sokratis Papastathopoulos doubled the hosts' lead shortly before half-time, lashing high into the net from close range after United failed to clear their lines at a corner, but Arsenal had capitulated from a winning position on this ground against Chelsea three days earlier.
There was to be no turnaround this time, however. The win moves Arsenal up to 10th in the Premier League table, while United remain fifth, but the balance of play suggested the two sides could end the season much closer together.
Chelsea held by Jahanbakhsh stunner
Cesar Azpilicueta scored the first Premier League goal of the decade with a close-range finish after Tammy Abraham's shot was blocked by Aaron Mooy, but Chelsea could not get the job done against Brighton.
Jahanbakhsh celebrated with tears of joy after scoring his first goal for Brighton in the 2-0 win over Bournemouth on Saturday and he did not have to wait long for his second goal.
The Iran winger sent an acrobatic attempt past Kepa Arrizabalaga with six minutes of normal time remaining to deny Frank Lampard's side a third away league win on the spin.
Foxes march on with win at 10-man Newcastle
Despite being without the Premier League's leading scorer Jamie Vardy for a second game, firstly due to the birth of his child and this time due to a minor injury, Leicester were comfortable 3-0 winners at Newcastle.
The Foxes' incredible campaign continued thanks to goals from former Newcastle attacker Ayoze Perez, James Maddison and substitute Hamza Choudhury.
While Leicester's outside title hopes just about remain intact, moving back to within 10 points of Liverpool after playing two games more than the runaway leaders, a third defeat in a row for Newcastle leaves them five points above the relegation zone.
Villa pick up much-needed win while Wickham pegs bag Norwich
Villa boss Dean Smith insisted he still had the support of the dressing room in the build-up to the trip to Burnley, and that showed as Villa returned to winning ways with a 2-1 triumph at Turf Moor that saw them climb out of the relegation zone.
Wesley got the first goal after 27 minutes and the Villans added a second on the stroke of half-time when Jack Grealish scored his sixth Premier League goal in his 19th appearance this season.
Chris Wood pulled one back from Burnley's first shot on target in the 80th minute, though it was not enough to prevent a third successive loss for Sean Dyche's out-of-form side.
Meanwhile, Todd Cantwell put Norwich City ahead after just four minutes of their clash with Palace, who battled bravely to stay in the game before substitute Connor Wickham turned Wilfried Zaha's cross into the net five minutes from time to earn his side a share of the points.
Kane limps off for stuttering Spurs, Watford's revival continues
Danny Ings scored the 16th goal Tottenham have conceded in their last 11 matches when he lifted the ball over Toby Alderweireld and fired past Paulo Gazzaniga for what proved to be the only goal of the game after 17 minutes at St. Mary's.
Tottenham have taken only four points from the 12 on offer over the festive period and their misery was compounded by Harry Kane limping off after hurting his hamstring, and Jose Mourinho was shown a yellow card by referee Mike Dean late on after being involved in touchline confrontation.
Watford maintained their impressive form with a 2-1 win at home to Wolves after Gerard Deulofeu opened the scoring half an hour in with a curled finish into the bottom corner.
Abdoulaye Doucoure fired in on his 27th birthday to double the home side's tally, but Pedro Neto reduced the deficit with a deflected shot on the hour and Christian Kabasele was sent off with 19 minutes to play.
Wolves could not find a way past Ben Foster again, though, meaning just a third loss in 16 league outings for Nuno Espirito Santo's men.