There were also important victories for Newcastle United and Fulham against Cardiff City and Stoke City respectively, while Hull City drew with Aston Villa.
With Arsenal not in action until Sunday, Liverpool took the opportunity to climb to the summit with a comfortable 3-1 over struggling Crystal Palace.
Making his first appearance at Anfield since April, Luis Suarez broke the deadlock in the 13th minute when he exchanged passes with Jose Enrique and, despite slipping to the floor, showed great improvisation to drag a shot past Julian Speroni.
Speroni was beaten again just four minutes later as Daniel Sturridge bamboozled Damien Delaney with a twisting run before driving a powerful strike inside the far post from a narrow angle.
Steven Gerrard got in on the act from the penalty spot with his 99th Premier League goal in the 38th minute after Raheem Sterling was fouled just inside the area by Dean Moxey.
Palace's fans were given something to celebrate on 77 minutes thanks to Dwight Gayle's glancing header from a Jose Campana free-kick, but it never looked like being anything other than a consolation goal as Liverpool cruised to victory.
Belgian teenager Januzaj came to Manchester United's rescue with a dazzling breakthrough performance as the Premier League champions hit back from a sorry start at the Stadium of Light to prevail 2-1 and leave Sunderland bottom of the table.
Shambolic defending by United in the fifth minute allowed Sunderland to take the lead as Craig Gardner took advantage of Nemanaj Vidic's failed clearance to score past David de Gea.
The Black Cats carved out further opportunities to extend their lead as Kevin Ball did his prospects of taking on the manager's job on a long-term basis no harm, but Januzaj dramatically turned the game in United's favour with two quick-fire goals in the second half.
The 18-year-old equalised in the 55th minute with an excellent finish from Patrice Evra's left-wing cross and then struck for a second time on 61 minutes as he volleyed exquisitely into the bottom corner.
Manchester City came from behind in the day's early game to beat Everton 3-1 and inflict a first Premier League defeat of the season on Roberto Martinez's men.
Romelu Lukaku kept up his fine form to cut inside Joleon Lescott and fire Everton in front with a firm low strike after just 16 minutes, although Joe Hart will have been disappointed not to have got a stronger hard on the ball after being criticised for his performance against Bayern Munich in midweek
Everton's lead lasted barely a minute before Alvaro Negredo's run was picked out by a threaded pass from Yaya Toure and he slotted past the onrushing Tim Howard.
Sergio Aguero spurned a couple of glorious chances but made amends on the stroke of half-time by latching on to David Silva's through-ball and finishing from a tight angle.
City took control in the second half and scored again from the penalty spot in the 69th minute after Pablo Zabaleta was brought down by Seamus Coleman, with Aguero's spot-kick rebounding off the post and going back in off Howard for an own goal.
Ease the pressure
Loic Remy scored twice as Newcastle United returned to winning ways with a 2-1 success at Cardiff City to ease the pressure on manager Alan Pardew.
Remy opened the scoring on the half-hour mark as he drifted in from the left and drilled a precise effort from 20 yards which David Marshall perhaps could have prevented from going in at his right-hand post.
Marshall did manage to save from Papiss Cisse eight minutes later after a defence-splitting pass from Yohan Cabaye, but Remy swept in the rebound majestically from the edge of the area to make it 2-0.
Peter Odemwingie reduced Cardiff's arrears on 58 minutes after collecting the ball from substitute Jordon Mutch and coolly sliding home, but Newcastle clung on to pull clear of trouble.
Mark Hughes endured an unhappy return to Craven Cottage as Darren Bent's late goal secured a vital 1-0 victory for Fulham against Stoke City.
Robert Huth had a header cleared off the line in the first half but neither goalkeeper was worked too hard and it looked like an uninspiring game would end goalless.
However, Bent was played in by Steve Sidwell with just seven minutes to go and wriggled into enough space to guide the ball home and give beleaguered boss Martin Jol cause for optimism.
In the day's remaining game, Hull City and Aston Villa collected a point apiece after playing out a goalless draw at the KC Stadium.
Both sides came into the match on the back of good results but there was nothing to separate them on a frustrating afternoon, with Gabriel Agbonlahor denied by a smart save from Allan McGregor and firing another good chance wide.