Liverpool returned to the top of the English Premier League table on Saturday after a 3-1 home win over Crystal Palace, while Fulham grabbed a late winner at home against Stoke City.
Luis Suarez was making his first Anfield appearance since completing a ten-game ban for biting Chelsea's Branislav Ivanovic last season and with new-born son Benjamin in attendance, he broke the deadlock in the 14th minute.
The goal showcased the Uruguayan's flair for improvisation, as he hooked the ball into the net from a prone position after slipping as he shaped to shoot.
Top scorer Daniel Sturridge added a second three minutes later, turning Damien Delaney inside out before lashing home his fifth league goal of the season.
Captain Steven Gerrard made it 3-0 with a 38th-minute penalty, taking his tally of Premier League goals to 99, before substitute Dwight Gayle replied for the visitors with a late glancing header.
Victory took Brendan Rodgers's side a point clear of Arsenal, who can reclaim top spot if they win at West Bromwich Albion on Sunday.
Man City back on track, Fulham grab late winner
Earlier, City bounced back from defeats by Aston Villa and Bayern Munich to end Everton's unbeaten start to the season with a 3-1 win at the Etihad Stadium.
Everton took a 16th-minute lead through on-loan Chelsea striker Romelu Lukaku, who broke the offside trap to collect Phil Jagielka's pass and deftly nutmegged Joleon Lescott before beating Joe Hart with a low shot.
It was his third goal in two games, following his brace against Newcastle United on Monday, but City replied within two minutes, Alvaro Negredo squeezing the ball past Tim Howard from Yaya Toure's fine reverse pass.
City were dealt a blow in the 34th minute when captain Vincent Kompany went off with an injury, but they moved ahead on the stroke of half-time.
David Silva strode forward and threaded a pass down the inside-right channel for Sergio Aguero, who rattled a low shot into the bottom-left corner.
A foul by Seamus Coleman on Pablo Zabaleta gave Aguero a chance to extend City's lead from the penalty spot in the 69th minute and although Howard turned his shot onto the post, the ball bounced against the American goalkeeper and into the net for an own goal.
"Equalising immediately was important, but more important was the character of the team, playing against Everton -- the only unbeaten team in the league," City manager Manuel Pellegrini said.
"After a defeat (by Bayern) on Wednesday, it was important for me to see how the players reacted."
Loic Remy scored twice in the space of eight first-half minutes as Newcastle United claimed a first win in three games with a 2-1 victory at Cardiff City, who replied through Peter Odemwingie.
Fulham climbed out of the relegation zone after a neat 83rd-minute strike from Darren Bent secured a 1-0 win at home to Stoke City, while Aston Villa were held to a 0-0 draw at Hull City.