Playing in front of the Kop for the first time since his 10-match suspension, Suarez received a warm welcome as he entered the pitch and gratified the Reds fans with a solid performance and a classy goal, finding the net after 13 minutes, surrounded by defenders after losing his balance.
Liverpool moved one point clear of Arsenal, which has a match in hand and travels to West Bromwich Albion on Sunday.
Defending champion Manchester United struggled at last-place Sunderland and came from behind to claim a scrappy 2-1 win as teenager Adnan Januzaj scored twice on his full debut.
The win, which lifted United to ninth place, will take some pressure off manager David Moyes after consecutive league defeats against City and West Brom.
Suarez's strike partnership with Daniel Sturridge again proved lethal, with the England striker doubling his team's lead four minutes later with a solo effort. The attacking duo have 12 goals between them in nine Premier League appearances.
"They were exceptional. Their combination play was outstanding and they are right up there with the best in this league," Liverpool boss Brendan Rodgers said.
Liverpool captain Steven Gerrard scored from the spot in the 38th for his 99th Premier League goal before Dwight Gayle headed home a consolation goal.
After two consecutive defeats in all competitions against Aston Villa and Bayern Munich, City fans were expecting a reaction from their team, and the players did not let them down as they defeated Everton 3-1 in an entertaining game featuring plenty of chances.
In other matches, Cardiff lost 2-1 at home against Newcastle, Fulham beat Stoke 1-0 and Hull was held to a 0-0 draw by Aston Villa.
Romelu Lukaku gave Everton the lead but Alvaro Negredo quickly cancelled out the Belgian striker's opener after a flowing collective move, before Sergio Aguero handed City the advantage just before halftime.
Aguero then added the third from the spot in the 69th minute after Tim Howard tipped his penalty onto the post but the ball bounced in off the goalkeeper.
Under pressure before the match following a string of below-par performances, Joe Hart did not make any blunders and got his hand at Lukaku's strike but could not prevent the ball from finishing in the net.
"Always the responsibility for the goal is with the whole team," Manchester City coach Manuel Pellegrini said. "Of course he feels he didn't do well the last game, he feels that he was guilty, but it was more important that today he reflected how strong his character is."
At Anfield, Mamadou Sakho missed the chance to extend his team's lead when he headed wide after the interval before Sturridge hit the post and Palace slumped to a fourth consecutive league defeat.
"I have been given an opportunity to play in a position I've played all my life," Sturridge said. "The manager has given me a chance at Liverpool and I thank him for giving me the opportunity to show the world what I can do."
While the Sturridge-Suarez partnership looked potent and well-oiled, United forwards Wayne Rooney and Robin van Persie failed to impress, like the rest of the team.
Sunderland took the lead in the fifth minute as Craig Gardner made the most of the Red Devils' poor defending but the 18-year-old Januzaj equalized early in the second half with a clean finish from Patrice Evra's cross before smashing a wonderful volley in the 61st.
In South London, Darren Bent scored a late winner to give Fulham a much-needed 1-0 victory over Stoke.
Fulham's first home win of the season, and only second overall after its opening-day victory at Sunderland, should relieve some of the pressure on under-fire manager Martin Jol.
At Cardiff, Loic Remy opened the scoring in the 30th with a solo effort as he broke down the left flank, then cut inside past two defenders and slotted a shot from outside the area inside the near post. He added the second with a deft finish from outside the area when a rebound fell into his path seven minutes later.
Peter Odemwingie pulled a goal back for Cardiff in the 58th but the Welsh club's comeback threat petered out.