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 were 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 half-time.
Aguero then added the third from the spot in the 69th minute after Tim Howard tipped his penalty on to the post but the ball bounced in off the goalkeeper.
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.
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 equalised 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.
At Cardiff, Loic Remy opened the scoring in the 30th, and added the second seven minutes later. Peter Odemwingie pulled a goal back for Cardiff in the 58th.
- AP