Liverpool showed their credentials in targeting a first title since 1990, and such was the Reds' dominance in their 2-1 win at Tottenham, the scoreline did not do Jurgen Klopp's men justice.
Chelsea also made it five wins from as many games under Maurizio Sarri thanks to the brilliance of Eden Hazard, who scored a hat-trick in a convincing 4-1 win over Cardiff to edge ahead of Liverpool on goal difference.
City remain just two points back, though, and looked back to their best in tearing apart Fulham 3-0 thanks to goals from Leroy Sane, David Silva and Raheem Sterling.
Manchester United and Arsenal stay six points off the top after 2-1 wins at Watford and Newcastle respectively.
Liverpool's first win away at Tottenham in four attempts for Klopp was another sign of the progress made under the German, particularly as they suffered a 4-1 thrashing at Wembley last season.
"Nobody will play the perfect season. It's still early, five games, fantastic we won all five and improved. I like that development," said Klopp.
"Now we have to prove that and do it again and again."
Spurs paid a heavy price for the absence of captain Hugo Lloris through injury as stand-in goalkeeper Michel Vorm flapped at a corner six minutes before the break and Georginio Wijnaldum looped a header over the line before Vorm could scramble clear.
Roberto Firmino tapped home a second nine minutes into the second half and Liverpool should have added to their lead as Sadio Mane and Naby Keita were denied by Vorm.
However, the visitors had to see out a nervy final few seconds when Erik Lamela skilfully volleyed home from a narrow angle and Son Heung-min had a penalty appeal waved away with the final attack of the game.
Sane returns in style
City and Chelsea await in Liverpool's next three league games and secured comfortable home victories to boost their own title chances.
Pep Guardiola started Leroy Sane for the first time this campaign and last season's Premier League Young Player of the Year looked more like his old self as Fulham were put to the sword.
Sane tapped home Fernandinho's inviting cross inside two minutes before Silva smashed his 50th Premier League goal and Sterling completed the rout.
"In the period he didn't play, we never doubted Leroy's quality," said Guardiola.
"He is 22 and this little bit of up and down can sometimes happen at this age. But I'm delighted with his performance, a big compliment to him."
Sarri is still to taste anything but victory in English football, but had a scare early on at home to Cardiff when Sol Bamba put the visitors in front at Stamford Bridge.
However, Cardiff's failure to significantly strengthen over the summer means they are liable to being outclassed at this level and were undone by the brilliance of Hazard.
"I thought that Hazard was one of the best players in Europe. But now I change my mind because maybe he is the best," said Sarri.
The Belgian scored twice in seven minutes to send Chelsea in ahead at half-time and completed his hat-trick from the penalty spot before Willian added a fourth.
United further eased the pressure on Jose Mourinho by ending Watford's 100 percent record at Vicarage Road.
Two goals in three first-half minutes from Romelu Lukaku and Chris Smalling put the visitors in command.
But they had to withstand some late pressure after Andre Gray halved their advantage and Nemanja Matic saw a late red card.
"In the second half we decreased our intensity, then the goal happened, then the game is open," said Mourinho.
"I think this last Lukaku tackle as a right-back shows everything. I think he is the image of the team, a team that played very well in long periods in the first half."
Mesut Ozil quietened his critics with his first Arsenal goal since February at St James' Park as the Gunners ground out a third straight win.
Bournemouth moved up to fifth with an impressive 4-2 victory over 10-man Leicester with Ryan Fraser scoring twice.
And Wilfried Zaha underlined his importance to Crystal Palace with the only goal on his return from injury in a 1-0 win at Huddersfield.