Mohamed Salah set up Sadio Mane's 100th Premier League goal and scored a brilliant goal of his own, while Roberto Firmino bagged a hat-trick as Liverpool reclaimed top spot with a 5-0 win over Watford in Claudio Ranieri's first game in charge.
The visitors took the lead in the ninth minute when Liverpool won a Watford kickout, and Mohamed Salah played a sublime pass to Mane with the outside of his left boot for the Senegalese striker to notch his Premier League century.
Mane then picked out James Milner on an overlapping run and he drove a perfect ball across the goal for Firmino to steer home from close range in the 37th minute to put the Reds 2-0 up at halftime.
Firmino added his second seven minutes after the break, pouncing on a defensive mistake to poke home, before Salah stole the headlines two minutes later with an outrageous display of close control, outwitting three defenders in a tight space before curling the ball into the net.
Ranieri, who took over as Watford boss on 4 October, following the sacking of Xisco Munoz, was left in no doubt about the size of the task he faces as Watford came up woefully short, and Firmino added his third in second-half stoppage time.
Chelsea kept their grip on the top spot in the Premier League as they beat neighbours Brentford 1-0 thanks to a Ben Chilwell strike, but only after a string of last-gasp defending and saves by goalkeeper Edouard Mendy denied the hosts.
Chilwell scored for the second match in a row in the 45th minute when Brentford failed to clear and the ball dropped kindly for the England defender to strike from 12 yards with a half volley.
The hosts had earlier hit the post through Bryan Mbeumo and the Frenchman was again denied by the woodwork in the 75th minutes as the Bees subjected Chelsea to wave after wave of attacks in their push for an equaliser.
Brentford spent much of the first half pegged back by Chelsea but the red and white shirts piled forward after the break, roared on by their supporters.
In the 83rd minute, Chelsea goalkeeper Mendy denied Saman Ghoddos and Trevoh Chalobah then saved the visitors by clearing off line from Christian Norgaard. Mendy made further fine saves, including turning over an acrobatic overhead effort by Norgaard in injury time.
Wolves came from two down to win 3-2 at Aston Villa with all five goals coming in the second half of a thrilling Midlands derby.
Midfielder Ruben Neves' 95th-minute free-kick deflected off Matt Targett to give Wolves the win that sent their fans into raptures but silenced most of the 41,961 crowd at Villa Park.
After a tight first half with chances at both ends, the hosts cruised into a 2-0 lead in the second period.
Villa striker Danny Ings scored his first goal since August with a powerful close-range header from John McGinn's cross in the 48th minute. Twenty minutes later, McGinn saw his low strike deflect off Neves into the net.
But Wolves clawed themselves back with goals from Romain Saiss, who swept in an 80th minute cross, and then from fellow defender Conor Coady, who got the last touch on a goalmouth scramble five minutes later.
With 2-2 looking a fair score, Neves settled the game in stoppage time, leaving Villa shell-shocked.
Bernardo Silva and Kevin De Bruyne were on target as Manchester City beat Burnley 2-0 at the Etihad Stadium.
Portuguese midfielder Silva opened the scoring in the 12th minute, tapping in after Burnley keeper Nick Pope had pushed out a low drive from Phil Foden.
Burnley caused City some problems however and Maxwel Cornet could have brought them level when he latched on to a ball over the top from Dwight McNeil but City keeper Zack Steffen did well to save with his legs.
Josh Brownhill then screwed a shot wide from inside the box after some hesitant defending from City as Burnley ended the first half strongly.
Riyad Mahrez went close to doubling City's lead after the break, clipping the cross-bar after a smart pull-back from Silva.
City finally put the contest to bed, in the 70th minute, when Ashley Westwood failed to clear for Burnley and the ball fell to De Bruyne who smashed home with his left foot.
Southampton claimed their first Premier League victory of the season as Armando Broja's second-half goal gave them a 1-0 win at home to Leeds United.
The three points mean Ralph Hasenhuettl's side rise to 15th in the table, while Leeds are one point and two places behind.
The hosts came close to opening the scoring early on when Broja flicked on a Nathan Redmond corner towards the back post but Mohamed Elyounoussi failed to meet it.
Redmond and Broja combined to break the deadlock in the second half when the former squared it to the on-loan Chelsea forward who netted his first Premier League goal, side-footing the ball into the roof of the net.
Dan James should have levelled for the visitors when he poked wide after a lapse from the hosts' defence, but that was as close as Leeds would come.
Despite a late surge in intensity, Marcelo Bielsa's men failed to create any real chances.
Leicester City ended their four-match winless run in the Premier League as goals from Youri Tielemans, Caglar Soyuncu, Jamie Vardy and Patson Daka helped them to a comeback 4-2 home win over Manchester United.
With the score 2-2 late in the game, Vardy netted a superb goal with the outside of his foot to cap a frantic five-minute spell before substitute Patson Daka put the icing on the cake for Leicester in stoppage time.
Mason Greenwood had fired United into a 19th-minute lead when he unleashed a thunderbolt from 20 metres that flew into the far corner as it shaved the inside of the post.
But the visitors were punished for sloppy passing in the danger zone in the 31st as Kelechi Iheanacho robbed Harry Maguire of the ball and squared it for Tielemans to hit the top corner with a looping shot over goalkeeper David De Gea.
A tame second half came to life when Soyuncu turned the tide with a close-range effort in the 78th minute before Marcus Rashford, returning after shoulder surgery in August, levelled for United in the 82nd.
Less than 60 seconds later, however, substitute Ayoze Perez chipped the ball into Vardy's path and the striker gave De Gea no chance, before Daka took advantage of more poor defending by United to tap the ball in at the far post.
Norwich City are still searching for their first Premier League win this season after playing out a goalless draw with Brighton & Hove Albion at Carrow Road.
Brighton dominated possession and had a penalty appeal denied in the first half when Neal Maupay claimed contact from goalkeeper Tim Krul, but Norwich squandered a golden opportunity towards the end of the half to take the lead.
Josh Sargent looked set to fire into an open net from outside the box when visiting goalkeeper Robert Sanchez raced out and missed a clearance, but the American forward's tame shot was cleared with ease by Shane Duffy.
Norwich remain rooted to the bottom of the league standings on two points.