Leon Osman and Steven Naismith were on the score sheet to stun Brendan Rogers' men.
Leon Osman tapped in Everton's fourth from close range after Johnny Heitinga's free-kick came off the wall.
Leon Osman put the visitors ahead when he headed in Landon Donovan's corner.
But they responded well and Steven Pienaar converted Leon Osman's cross with a fine strike to open the scoring.
Leon Osman gave the home side a half-time lead only for Darren Bent to equalise straight after the restart.
But they went ahead when Blues defender Roger Johnson could only prod into his own net from a Leon Osman cross.
Leon Osman's feint created some space on the edge of the box before his shot was snuffed out by a crowd of defenders.
Everton's Leon Osman and Richard Dunne may seem an odd couple but they've brought a little warmth back into a cold, depressing few days.
First half goals from Kevin Mirallas and England international Leighton Baines put Everton in control before Leon Osman made it 3-0 on the hour mark.
Everton finally hit back in injury-time as striker Louis Saha, who replaced midfielder Leon Osman in the second half, netted after Steven Pienaar's shot was blocked.
Midfielders Steven Pienaar (hamstring) and Leon Osman (stomach strain) have both missed recent games, while Anichebe was injured by a horror tackle from Kevin Nolan against Newcastle.
Everton were boosted by the return of Tim Cahill after suspension and Leon Osman after injury - and David Moyes's side took charge in the first 45 minutes.
Everton managed to weather that storm and carved out a couple of their own openings, Leon Osman heading straight at Petr Cech and Beckford skewing way off target.
The duo were involved in the build-up as Leon Osman struck the outside of Marton Fulop's left-hand post with a sweetly struck, right-foot curling effort from the edge of the area.
Baines then crossed for Leon Osman but his effort from 10 yards was deflected wide, while Eddie Nolan made a superb block to thwart Saha after Fellaini's backheel freed Baines to centre from the left.
The Toffees carved the game's first opening as Louis Saha exchanged passes with Cahill and fired a shot which was smothered by Rui Patricio, and the Sporting goalkeeper was alert enough to block Leon Osman's follow-up effort.
Even so, goals from midfielder Leon Osman and substitute Nikica Jelavic kept David Moyes' team in sixth place, three points ahead of city rivals Liverpool with a game in hand as the club's manager celebrated 11 years in charge.
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Everton, with Leon Osman the game's outstanding performer, were afforded a standing ovation by their ecstatic supporters at the final whistle - and rightly so after they demonstrated the confidence to match, and better, United in all areas of the pitch.
Former City defender Sylvain Distin headed Everton's equaliser from Mikel Arteta's free-kick after 65 minutes with goalkeeper Joe Hart at fault - and as Roberto Mancini's men rocked under increasing pressure, Leon Osman rose spectacularly to beat City captain Vincent Kompany and head the winner seven minutes later.
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