With top scorer Sergio Aguero facing at least a month out, Dzeko took his chance to shine at the King Power Stadium by netting twice after Aleksandar Kolarov opened the scoring with a free-kick.
Leicester, third in the Sky Bet Championship, had gone into the game with confidence but were outclassed by a City side showing eight changes but finished strongly and claimed a deflected Lloyd Dyer consolation.
Even though Manuel Pellegrini rotated his squad, the City side he fielded was still too powerful for the Foxes.
Vincent Kompany, David Silva and Pablo Zabaleta were the players retained from the weekend while Joleon Lescott and James Milner were among those to come in with Hart.
The only obvious downside for City, who now have semi-finals to look forward to in January, was an injury to Zabaleta.
The Argentinian limped off with what appeared a hamstring problem early on to join fellow right-back Micah Richards on the casualty list.
Pellegrini could now have a problem in the position for a Christmas programme that includes a home clash with Liverpool on Boxing Day.
Leicester could at least take heart from the way they finished ahead of their crunch clash with QPR.
It soon became apparent that the Premier League side, for all Leicester's enthusiasm, were a cut above, even at half-pace.
Kasper Schmeichel was exposed all too easily and but for the Leicester keeper, against one of his father Peter's former clubs, the visitors may have inflicted more damage. Jack Rodwell first tested him from inside the area after combining nicely with Dzeko.
Schmeichel could do little to keep out Kolarov's superb free-kick after eight minutes, the Serbian curling into the top corner from 30 yards after City had broken from deep.
Leicester responded well but were unable to find the final ball.
One promising break was ended crudely when Jesus Navas bundled over Dyer but generally City, well marshalled by Kompany, looked to have ironed out recent vulnerabilities.
Gradually the chances began to come for City as Dzeko had a shot blocked, Rodwell screwed wide and a fierce Kolarov drive was punched behind by Schmeichel.
Kompany had a header from a corner blocked, Rodwell curled another shot wide and Silva also missed the target.
Dzeko went very close to a second as he turned outside the area and curled a left-foot shot millimetres wide of the far post.
The Bosnian finally got on the scoresheet five minutes before the break as headed past Schmeichel from a well-weighted Milner cross.
Milner and Navas then both brought further saves out of Schmeichel before the interval.
Schmeichel was in action again, saving from Dzeko within moments of the restart but he could not prevent him scoring again for long.
Milner again cut through Leicester on the left and Dzeko turned in his cut-back at the near post past an unsighted Schmeichel, who made his anger with the defence clear.
At the other end Hart had hardly been troubled, although he did need to keep out one Andy King effort.
Silva then tested Schmeichel again before leaving the field, but things hardly threatened to ease up for Leicester as Samir Nasri took his place.
Yet Leicester did manage to finish with a flourish and Dyer took in a ball from Paul Konchesky and lashed in a shot which took a wicked deflection off Kompany and beat a wrong-footed Hart.
Substitutes Matty James and Jeff Schlupp also gave Hart some shots to save but Leicester were unable to set up a frantic finale.