The Welsh side created several attacking opportunities and were comfortably the more dangerous team, yet Napoli will return home as favourites to progress following next Thursday's last-32 return leg.
Rafael Benitez's men were fortunate to escape from south Wales with a 0-0 draw, such was the quality of Swansea's display.
The Serie A outfit, who were only eliminated from this season's Champions League group stages on goal difference, had their hands full coping with striker Wilfried Bony's power and winger Nathan Dyer's trickery.
Ashley Williams twice had headers saved after coasting in behind Napoli's defence, while Swansea were always ready to try their luck from long range, meaning the visitors could never relax.
It was just the fourth game of new Swansea boss Garry Monk's reign, but the signs were all positive that he will continue the impressive work of his predecessors Michael Laudrup, Brendan Rodgers and Roberto Martinez.
Ultimately, though, it was a night of frustration for the Swans after arguably a season's best performance did not bring the outcome such dominance warranted.
Monk, taking charge of his first European match since succeeding Laudrup earlier this month, made eight changes from the side beaten by FA Cup fifth-round opponents Everton last weekend, including returns for Williams, Chico Flores and Leon Britton.
Napoli's on-loan Liverpool goalkeeper Pepe Reina featured on the bench for Benitez's side, but top-scorer Gonzalo Higuain started after shaking off a knock.
The Serie A side served immediate notice of their quality by threatening Swansea's goal twice inside the first five minutes.
Jose Callejon arrowed a shot narroly wide, then Swansea goalkeeper Michel Vorm was forced to save low down from Napoli skipper Marek Hamsik before Swansea hit back through an audacious Dyer attempt that Rafael Cabral tipped around the post.
Both teams were prepared to attack with purpose and quality, and there was no let-up in the action as Bony blasted a long-range shot narrowly wide.
Napoli knew they were in a game, and Bony went close to opening the scoring midway through a breathless first-half after Pablo Hernandez's defence-splitting pass sent him clear on goal, only for Cabral to pull off a stunning save at the Ivorian striker's feet.
Swansea continued to dictate the tempo, with Bony a consistent threat to Napoli's defence and Dyer causing repeated problems with his pace and close control, but Cabral remained unbeaten as half-time approached.
Williams should have opened the scoring six minutes before the break, but his angled header flew over the bar from close range, much to Cabral's relief.
Cabral was then forced into making two quality saves in as many minutes, first of all palming away Wayne Routledge's shot before deflecting an Angel Rangel header clear of danger.
He required treatment following the second stop, which resulted in Reina warming up on the sidelines, and the half-time whistle came as welcome relief to the visitors after a scoreless opening period that had seen them outplayed.
Cabral did not appear for the second-half, which meant Reina taking over and hoping he would not be pressed into as much action as his team-mate had been in thwarting a number of Swansea attempts on goal.
Monk made his first substitution after 56 minutes, sending on Jonathan De Guzman for Hernandez, whose last contribution was to be booked for a challenge, and then Napoli's Lorenzo Insigne received a yellow card after he lunged at Dyer.
Swansea, after their first quiet spell of the game, tested Reina when De Guzman's floated free-kick found Williams, but his header was saved at point-blank range.
Hamsik and Christian Maggio took Napoli's yellow card count to three, while Monk sent on Jonjo Shelvey for his first taste of action since New Year's Day, but the deadlock could not be broken and Napoli moved into the box-seat for next week's return.