Gabriel Jesus, Riyad Mahrez and Raheem Sterling were on target as champions Manchester City won 3-1 at Everton on Saturday to stay five points behind Premier League leaders Liverpool.
Pep Guardiola's side grabbed a 24th-minute lead when Brazilian Jesus stooped at the back post to head in a Kevin De Bruyne cross.
A determined Everton got back on level terms, though, nine minutes later, when after poor clearance from Fernandinho, Seamus Coleman lofted the ball goalwards and Dominic Calvert-Lewin dived to make sure with a header on the line.
City got back in front with a low free kick into the far corner from Mahrez after Yerry Mina had brought down De Bruyne.
Sterling made sure of the three points, blasting home off the underside of the bar after a low cross from substitute Sergio Aguero was deflected out off the foot of Everton keeper Jordan Pickford.
Tottenham defied Serge Aurier's first-half red card to hold out for a vital 2-1 win over Southampton, thanks to Harry Kane's winner.
Spurs had won just two of their opening eight games of the season and needed to make a statement after crashing out of the League Cup to fourth-tier Colchester on Tuesday.
Mauricio Pochettino's men got the early lift they desired from Tanguy Ndombele's opener. But they put themselves on the back foot as Aurier was booked twice in quick succession and saw red on 31 minutes. Then a huge error from Hugo Lloris handed Southampton an equaliser through Danny Ings.
Spurs showed their quality in a flowing counter-attack led by Son Heung-min and Christian Eriksen before Kane applied the finish touch to retake the lead before half-time.
The Champions League finalists demonstrated plenty of heart after the break to hold out for a precious three points that eases them back up to fourth in the Premier League table.
Manager Mauricio Pochettino said: "We have worked very hard for five years to put the club in the level that no-one expected. That is why now for some results that were not in the way people expected, we don't give up, so we need to be strong to be stronger."
On the Lloris blunder, Pochettino said: "When you're a professional player sometimes you take risks, if someone wants to blame Hugo for this, better to blame me because I demand him to play in this way."
Second-half goals from Jorginho with a penalty and Willian's deflected shot earned Frank Lampard's Chelsea a 2-0 victory over Brighton and Hove Albion, their first home Premier League win of the season.
Jorginho slotted the penalty into the bottom corner after Brighton defender Adam Webster had miscontrolled the ball and brought down Mason Mount as the young Chelsea midfielder rushed in to take advantage.
Willian hit home after a crossfield ball from substitute Callum Hudson-Odoi in the 76th minute. Goalkeeper Mat Ryan, who had held Chelsea at bay for most of the game, was wrongfooted by a deflection off Dan Burn.
Chelsea manager Frank Lampard, said: "We needed the win. We haven't had a 90 minute game where we've controlled it enough. The only disappointment is we didn't have more goals."
Aaron Cresswell fired in a late equaliser as West Ham stretched their unbeaten Premier League run to six matches.
Andriy Yarmolenko opened the scoring for the Hammers after 10 minutes with his third goal in six games. But the lead lasted just seven minutes as Joshua King swept the ball in and although it was initially flagged offside, the video assistant referee awarded the goal after a lengthy delay.
Callum Wilson swept in his fifth goal of the season straight after half-time only for Cresswell to deny the Cherries victory with a 74th-minute equaliser.
Wolverhampton Wanderers beat Watford 2-0 to claim their first win of the season thanks to a first-half strike by Matt Doherty and an own goal from Daryl Janmaat at Molineux.
Wolves started the day second-bottom of the table but defender Doherty began the process of lifting them out of the relegation places in the 18th minute, a simple tap-in from Pedro Neto's pass.
Doherty was also involved in his side's second on the hour mark, chipping a cross to Morgan Gibbs-White, who had only just come on, and his flick-on was headed into his own net by Janmaat.
Second-half goals from Jay Rodriguez and Chris Wood helped Burnley come from behind twice in a 2-2 with Aston Villa. John McGinn and Anwar El Ghazi scored for the home side at Villa Park.
Crystal Palace's Luka Milivojevic and Andros Townsend struck either side of half time for a deserved 2-0 home win over Norwich City.