The Swans, playing in their first ever knockout round in European competition, turned in an impressive display but just fell short at the San Paolo stadium.
Lorenzo Insigne put the hosts ahead in the 16th minute before Jonathan De Guzman equalised on the half-hour mark.
Gonzalo Higuain came to the rescue in the 78th minute to put Napoli back in front.
The Swans came agonisingly close to making it 2-2 with two minutes remaining but Napoli goalkeeper Jose Reina produced a fantastic one-handed save to repel Dwight Tiendalli's header.
Gokhan Inler put the tie beyond any doubt when he scored his team's third seconds from time and Napoli will take on Porto next.
Swansea coach Garry Monk was hoping for another strong performance from his players after their impressive display in the goalless first leg in Wales last week.
Swansea almost went ahead in the 11th minute when Pablo Hernandez fed Marvin Emnes on the right side of the box and his goal-bound shot beat Reina but was met with a goal-line clearance by Raul Albiol.
Napoli then wasted a good chance when Goran Pandev set up an unmarked Higuain in the box and his strike went over the bar.
Napoli broke the deadlock soon after when Higuain spotted Insigne and he beat a haphazard offside trap before lobbing the ball over Swansea goalkeeper Michel Vorm.
Swansea replied with De Guzman's stinging drive from 30 yards but it was saved by Reina.
At the other end, Higuain's right-footed shot from the edge of the area went just wide of Vorm's near post.
Midway through the first half, Ben Davies' dangerous corner-kick was cleared by a Napoli defender as the Swans looked for the equaliser.
Their efforts were rewarded when Wilfried Bony cleverly sent De Guzman through and the Dutchman surged forward before drilling an 18-yard shot past an onrushing Reina to send 900 Swans fans at the stadium into raptures.
The visitors grew in confidence and shortly after Hernandez forced Reina to dive to his right to clear his 20-yard attempt.
Reina then had to come off his line to clear a dangerous ball with Wayne Routledge ready to pounce.
Napoli tried to attack before half-time but could not find a way through.
De Guzman was first to test Reina in the second half with a 20-yard effort which was saved by the keeper.
Monk made his first substitution shortly after the hour mark, with Nathan Dyer replacing Routledge.
Swansea almost went ahead in the 66th minute when Bony met Hernandez's perfect cross but his header was saved by Reina.
Napoli continued to press forward and went ahead when Davies failed to clear the ball in the box and Higuain struck the ball past Vorm.
Reina then flew to clear Tiendalli's header from Hernandez's cross.
With Swansea forward looking for the equaliser, Napoli took advantage and substitute Marek Hamsik found Inler in the box and he fired in Napoli's third to seal the win.