In a thrilling encounter at Goodison Park, City did enough to claim three points and edge ahead of Liverpool by virtue of a significantly superior goal difference.
Manuel Pellegrini's men can therefore all but guarantee a second league crown in three seasons by winning their remaining matches, at home to Aston Villa and West Ham, while Arsenal are now assured of the final UEFA Champions League qualifying place as a result of Everton losing.
Ross Barkley opened the scoring for the hosts with a remarkable long-range effort, but Everton boss Roberto Martinez was made to pay for an ambitious 3-4-3 formation as City exploited his team's left-hand side.
England World Cup hopeful Barkley brought Goodison Park to its feet by curling Steven Naismith's lay-off into the top corner from 25 yards.
Perhaps inspired by their captain's heroics, City drew level in the 21st minute.
Howard thwarted Dzeko from an acute angle but when the ball was worked out to James Milner the big Bosnian was alive to the possibilities and headed home from his team-mate's pinpoint cross.
In a similar response to the one following Kompany's earlier brilliance, City streamed up field to give themselves breathing space.
Fernandinho sublimely controlled the attack and found Samir Nasri, whose low centre was turned in by Dzeko for City's 150th goal in all competitions this term.
Faced with the seemingly routine prospect of closing out a decisive win, City then faded badly and Lukaku reduced the arrears when he stooped to convert Leighton Baines' arching cross.
Bizarrely, City's defenders were then of greater use at the other end of the field, as Zabaleta and Demichelis went close to adding a fourth.
Lively Everton substitute Gerard Deulofeu rippled the side netting after cutting through the visitors with a mazy run.
