Mauro Icardi's late penalty earned Inter a point as Rafael Benitez's men missed the chance to close the gap on second-placed Roma, who drew with Chievo earlier in the day.
Marek Hamsik's 51st-minute header and a superb long-range strike from Gonzalo Higuain put Napoli in control.
However, Rodrigo Palacio pulled a goal back with 18 minutes remaining and the turnaround was completed in the 87th minute courtesy of Icardi's chipped penalty after Henrique received his second yellow card for a tug on Fredy Guarin.
Inter coach Mancini has urged his players to be more streetwise and was left to reflect on what might have.
He said: "Although we were playing at Napoli against a great side, the disappointing thing is losing 2-0 when you shouldn't be in that situation.
"We might have to improve in terms of experience and become a bit more cynical. That said, Napoli have a lot of quality up top and can cause you problems at any moment.
He added: "I'm not happy because towards the end we should have tried to win the game. We should have pressed for a third goal."
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