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Inter struck back early in the second half when Dzeko profited from a fortuitous deflection to hammer home a finish, as Simone Inzaghi's side avoided a second damaging defeat in a row following last weekend's Milan derby loss.
The Nerazzurri's derby defeat opened up the title race and handed Napoli the chance to go top with a win in their bid for a first league title since the Diego Maradona-inspired 1989-90 season.
The hosts took an early lead when referee Daniele Doveri awarded a spot kick after a look at the pitchside VAR monitor, which captain Insigne dispatched.
Zielinski was inches away from doubling Napoli's lead with a curler from the edge of the box that struck the woodwork, while Denzel Dumfries dragged a shot well wide from a good position with Inter's best chance of the first half.
Inter levelled less than two minutes after the restart when Dzeko's header struck Giovanni Di Lorenzo's foot and bounced back into the path of the Bosnian, who drilled a finish off the bar and in from close range.
The visitors had captain Samir Handanovic to thank for keeping them level, as the Slovenian reacted sharply to stop an Osimhen strike before superbly charging down an Eljif Elmas effort at the back post.