With Rafael Benitez's side strongly tipped as one of the tournament's favourites, many expected the Serie A outfit to progress following a 0-0 draw in Wales last week in the first leg of the last 32 tie.
But for almost 50 minutes at the Stadio San Paolo it looked as if Jonathan de Guzman's goal would see Swansea through via the away goals rule after Lorenzo Insigne had earlier put Napoli ahead.
Insigne latched onto Higuain's headed flick in the 17th minute and lobbed a deft finish over the advancing Michel Vorm, but Swansea responded strongly and were level 13 minutes later.
A glorious team move eventually found De Guzman in space and the Dutchman fired a left-footed strike beyond Pepe Reina to put his side in the ascendancy.
The visitors looked assured for much of the game, but Swansea's hopes were dashed 12 minutes from time when an unfortunate ricochet off Ben Davies fell kindly for Higuain, who smashed the ball beyond a helpless Vorm.
Inler rubbed salt into the wound in the final minute of stoppage time, coolly slotting beyond Vorm as Swansea left themselves exposed in chasing a second equaliser.
Victory means Benitez, who won the tournament last year while in charge of Chelsea, remains on course to defend his title.
Marvin Emnes, one of three Swansea changes to the starting XI from the home leg, had the first sight of goal in the 11th minute, but, after being played in by Pablo Hernandez, the Dutchman saw his chipped effort over Reina cleared from just in front of the goalline by Raul Albiol.
The hosts ought to have broken the deadlock themselves four minutes later, but Higuain blazed over from 10 yards after linking up with Goran Pandev.
Swansea failed to heed the warning in the 17th minute, Insigne springing the offside trap before lofting his finish over Vorm and into the net.
The Swansea defence was in disarray, Higuain flashing an effort a yard wide with the visitors apparently still reeling from falling behind.
The Premier League side recovered their composure, however, and were level on the half-hour mark.
A slick passing move resulted in the ball being fed into Wilfried Bony and his clever flick was picked up by De Guzman, who fired left footed beyond Reina to provoke jeers from the home crowd.
The goal gave Swansea added confidence and they enjoyed long spells of possession, Bony cutting inside Henrique before testing Reina as the half drew to a close.
Midway through the second half, Insigne broke through only to see his effort blocked by Chico Flores, before Bony wasted a glorious chance when he headed Hernandez's cross straight into the arms of Reina.
Swansea hearts were broken on 78 minutes when the ball bounced off both Flores and Davies and landed perfectly for Higuain, who made no mistake from 10 yards.
With almost the last kick of the game and with Swansea completely exposed, substitute Marek Hamsik fed Inler, who beat Vorm to seal the win and set up a tie with Porto in the next round.
