Liverpool came through a tough Premier League test at Crystal Palace on Saturday with a 2-1 win at Selhurst Park.
Manchester City beat Chelsea 2-1 after coming from behind at the Etihad Stadium, with the champions climbing to third.
Jose Mourinho enjoyed victory during his first match in charge of Tottenham Hotspur, winning 3-2 at West Ham United.
A 96th-minute equaliser by Alexandre Lacazette rescued Arsenal for a 2-2 draw at home to Southampton.
Leicester's fine run gained more energy with a 2-0 away victory at Brighton & Hove Albion, and Everton continued to misfire after losing 2-0 against Norwich City at Goodison Park.
Elsewhere, Wolverhampton Wanderers won 2-1 at Bournemouth, and Burnley were comfortable 3-0 winners at Watford.
West Ham 2-3 Spurs
Bournemouth 1-2 Wolves
Arsenal 2-2 Southampton
Brighton 0-2 Leicester
Crystal Palace 1-2 Liverpool
Everton 0-2 Norwich
Watford 0-3 Burnley
Manchester City 2-1 Chelsea
Palace gave Liverpool the type of battle expected from Roy Hodgson's team, but the leaders once again passed their test with flying colours.
The hosts thought they had stolen the lead before half-time after James Tomkins struck, but the video assistant referee intervened to save Liverpool.
Liverpool turned the screw in the second half and took the lead through Sadio Mane just four minutes after the restart.
However, Palace did not falter, and the hosts were rewarded with an 82nd-minute equaliser from Wilfried Zaha.
Despite the endeavour from Hodgson's men, Liverpool went after the three points, and their mission was complete after Roberto Firmino passed the ball home with five minutes left.
The Reds have now gone 30 consecutive games without a Premier League defeat, and they are fully in control of their destiny.
City got their title challenge back on track after a comeback win against Chelsea at the Etihad.
The visitors had more possession than the hosts on the night and took the lead through N'Golo Kante's goal after 21 minutes.
Kevin De Bruyne gambled from the edge of the box eight minutes later, and the Belgian's shot deflected past Kepa Arrizabalaga.
Chelsea's defence were unable to contain Riyad Mahrez after 37 minutes, with the Algeria captain beating Emerson for pace to score.
The Blues provided a credible display but didn't have an answer in the second half, and City climb above their opponents.
Mahrez said after the game City were happy with the result:
Mourinho got off to the perfect start as the new Spurs boss after his team provided an impressive attacking display.
Dele Alli was deployed as the No. 10 behind Harry Kane, and the away team's progressive intent decided the game.
Heung-Min Son gave Spurs the lead after a quick breakaway down the left after 36 minutes, and Lucas Moura tapped home to double the score two minutes before half-time.
Kane made it 3-0 soon after the restart, and the game appeared beyond the Hammers long before the final whistle.
However, late goals by Michail Antonio and Angelo Ogbonna gave the hosts consolation, but Spurs were worthy winners at the end.
The success was Spurs' first Premier League away win for 10 months, and the new boss appeared satisfied with the result.
Arsenal's league performances continue to decline, and the Gunners were fortunate to claim a point against Southampton.
Saints were inspired at the Emirates Stadium, but they were consistently aided by Unai Emery's weak defence.
Danny Ings fired home the opener after eight minutes for the visitors. However, Lacazette made it 1-1 just 10 minutes later.
The game remained even until Kieran Tierney fouled Ings in the box, but James Ward-Prowse's 71st-minute penalty was saved by Bernd Leno, only for the Saints midfielder to rebound his effort home.
Arsenal struggled for the remainder of the match, and Southampton had multiple chances to finish the job.
However, deep into injury time, Lacazette broke Southampton hearts with an unexpected equaliser, finishing after Gabriel Martinelli's cross.