Dzeko, the Bosnia striker, was selected ahead of Alvaro Negredo by City boss Manuel Pellegrini for Tuesday night's tussle at Old Trafford and was on the scoresheet after just 43 seconds.
United's defence was in disarray as Fernandinho passed left for Samir Nasri, and although the Frenchman's shot hit the post, the lurking Dzeko netted from close range.
It took at least 15 minutes for the hosts to settle but even before the half-hour mark, both Danny Welbeck and Marouane Fellaini were somewhat fortunate to only receive yellow cards for challenges on Pablo Zabaleta - particularly the Belgian, who appeared to elbow the City full-back.
Juan Mata passed up a great chance to equalise towards the end of the first half, but it took just 11 minutes after the restart for Dzeko to score again for City.
He timed his run brilliantly to beat Rio Ferdinand and connect with a corner from Nasri, the ball flying past David de Gea.
United could not find a response and were punished even further when Yaya Toure snaked in a third in the 90th minute.
David Moyes' men slip 15 points behind their near neighbours in the table, after suffering a sixth home league defeat of the campaign. City now lie three points behind leaders Chelsea, with two games in hand.
Wilfried Bony handed the Swans an 11th minute lead, powering home Neil Taylor's cross.
But the Gunners hit back with two second half goals in the space of 66 seconds.
Substitute Lukas Podolski bagged his first Premier League goal of 2014 before setting up Olivier Giroud for his 19th goal of the season.
However, the visitors picked up a share of the spoils when Leon Britton's run caused havoc in the home defence and Mathieu Flamini put into his own net.
Arsene Wenger's side will now be looking over their shoulder with Tottenham and Everton breathing down their neck for fourth position.
Barkley produced a stunning solo run from inside his own half before powering into Newcastle territory and firing into the roof of Tim Krul's net.
Osman picked out Gerard Deulofeu on the right wing shortly after the restart and the Barcelona loanee slid an inch-perfect square ball for Lukaku to crash home.
The visitors added a third shortly before time as Deulofeu and Lukaku linked up to find Everton captain Osman, who struck into the top right-hand corner with power and precision.
A first away win of the calendar year sees Everton elevated to fifth in the Premier League standings, one point ahead of Tottenham, who they still have a game in hand on.