The moody Argentina forward, now the league's leading scorer with 12 goals, fired the hosts ahead after 64 seconds and Inter were in more trouble in the top-of-the-table clash when Yuto Nagatomo was sent off in the 44th minute.
The contest seemed to be over when Higuain struck again on the counter just after the hour but Inter were not finished.
Adem Ljajic quickly pulled a goal back in the 67th before Stevan Jovetic headed against the post and Miranda's header was somehow tipped on to the upright by Napoli keeper Pepe Reina in an unforgettable finale.
Earlier, Fiorentina, who had started the day in joint second with Napoli, were held 1-1 at Sassuolo.
Napoli, who took only two points from their opening three games, lead the table with 31 points from 14 games after winning nine and drawing two of their last 11 matches. Inter have 30 points and Fiorentina are third with 29.
The match in Naples was billed as a battle between Napoli's more refined style and Roberto Mancini's dour and physical Inter side.
Barely a minute had passed when Lorenzo Insigne caused havoc in the Inter penalty area and the ball fell to the feet of Higuain, who fired past Samir Handanovic with a shot on the turn. Jose Callejon and Marek Hamsik missed further chances for Napoli while Fredy Guarin and Ivan Perisic were close for the visitors.
Inter suffered a severe setback when Nagatomo was given a second booking for a clumsy, mistimed tackle on Allan.
The home side's second strike stemmed from an Inter goal kick. Napoli defender Raul Albiol headed Handanovic's punt straight back over the Inter defence to Higuain, who held off two defenders to blast the ball into the far corner.
Ljajic gave Inter hope when he replied five minutes later, managing to get in a shot despite being surrounded by three defenders, to deny Napoli an eighth successive clean sheet in all competitions.
Inter found a new lease of life, forced a succession of free kicks and corners and were desperately unlucky not to take a point as Jovetic headed against the post with Reina well beaten before Miranda was foiled by the keeper in successive attacks.
In the other game, Borja Valero gave Fiorentina a fifth minute lead at Sassuolo and Sergio Floccari headed the hosts level just before halftime.
(Writing by Brian Homewood; editing by Ken Ferris)
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