Manchester City were not just content with moving directly behind Chelsea at the top of the Premier League. Manuel Pellegrini's team played as though determined to make Manchester United suffer and by the time they had finished, David Moyes may have heard the first isolated shouts behind his dugout to get out of the club, and preferably through the back door.

For now it still amounted to a show of restraint considering he inherited a team, lest it be forgotten, that won the league by 11 points. The regression is nothing short of remarkable and the indignities continue to stack up. United certainly lived up to Sir Bobby Charlton's admission about having played "really, really badly" this season and Mancunian humour has a brutal edge on these occasions. "Worst champions we've ever seen" was the chant from the away end, expletive removed.

Moyes' new system, forfeiting width in an attempt to pin down City's midfield, failed within the first minute and Edin Dzeko's goal, officially clocked at 43 seconds, was followed by a performance of almost total superiority from Pellegrini's men, easing above Liverpool into second place with two games in hand on Chelsea and three points to make up.

City played like champions whereas United looked what they are these days: a team in seventh position, grubbing around for points to qualify for the Europa League, but in trouble as soon as they come up against one of the top teams. Dzeko's second goal arrived after 56 minutes and Moyes should probably be grateful the crowd did not let out more frustration after Yaya Touré completed the scoring.

United are now guaranteed their most undistinguished points total in the Premier League era and it is the first time City have beaten them three times running at this ground for more than 40 years. It is shocking to see the scale of deterioration at Old Trafford.

Pellegrini's men started like a team in a hurry, inflicted the necessary damage and coasted through the final half an hour before Touré drilled in the third goal just as the electronic board was flashing up the added time. They moved the ball quickly, with invention and clarity. There was not a flicker of trepidation and as soon as they took that early lead everyone must have sensed this could be another ordeal for the home team.

A blur of speed and movement culminated in Samir Nasri's shot ricocheting off a post for Dzeko to score from inside the six-yard area. United were all over the place and, straight away, there was the hard evidence about where these old rivals currently stand. The truth – and it is undeniable – is that the gulf in talent was evident all night.

More than anything, City had the speed to hurt the home team. Jesús Navas was a formidable opponent for the creaking Patrice Evra. On the opposite side, Rafael da Silva looked just as vulnerable. City simply overpowered the home team. David Silva was tremendous, such an elusive and clever player. Touré demonstrated why Pellegrini had described him beforehand as "the complete midfielder" and Dzeko played with the sureness of touch that occasionally deserts him.

The irony was that Moyes had shoehorned in another central midfielder, Tom Cleverley, at the expense of a right-winger to try to cope with City's superiority in this area. Still, however, Pellegrini's men poured through the gaps. Cleverley was removed at half-time and is in danger of becoming the player the crowd trust the least. Shinji Kagawa could not make any real difference and Marouane Fellaini continues to look hopelessly out of his depth. The player Moyes signed from Everton for £27m offered little in a passing sense and, not for the first time, the crowd revelled in his substitution.

As for the elbow Fellaini delivered into Pablo Zabaleta's jaw, half an hour in, it really was a dereliction of duty from the referee, Michael Oliver, not to show a red card. Fellaini has previous for these kind of offences and this one looked premeditated.

If City had been more clinical they could have wrapped up the game in that opening half. Wayne Rooney was determined not to let the game pass him by and United did at least create a couple of chances before the break.

Yet this was not a fully coherent football team. In defence there were more lapses than they will care to remember. United's midfield was laboured. They have scored 18 league goals here all season – even in the year of the infamous "Ta-ra, Fergie" banner, they managed 26 – and it does not reflect well on Moyes that Paul Scholes' analysis, in a television role, was much clearer than the manager's.

Early in the second half the outstanding Vincent Kompany headed on a corner and Fernandinho flashed the ball over the crossbar. There was a warning there for United and it went unheeded. At the next corner, Rio Ferdinand started tracking Dzeko but ran into Fellaini. Dzeko, left alone, ran on to Nasri's cross and expertly volleyed in his shot.

Touré's goal was a low diagonal drive into the bottom corner and that was the moment City's fans celebrated the Moyes era, in schadenfreude, again. "Five more years" they sang, toasting a night that could go a long way to determining whether they are reunited with the Premier League trophy.