Rookie Swans boss Garry Monk was up against UEFA Champions League winner Rafa Benitez, with the Welsh side comfortably the better side on the night, but unable to make the most of the chances they created.
Napoli, who dropped into the Europa League after finishing behind Borussia Dortmund and Arsenal in the Champions League despite claiming 12 points from their group stage campaign, will be favourites to progress with the second leg on home turf, but were fortunate to depart with a clean sheet.
Napoli carved out the first chance with only five minutes gone and Michel Vorm was called upon to produce arguably the best stop of the half, plunging away to his left to push a stinging low shot from Marek Hamsik wide of the target.
Swansea were a whisker away from taking the lead with only six minutes gone as Nathan Dyer's curling first-time effort from the edge of the area was clawed away by visiting goalkeeper Rafael.
And the shot-stopper, who was preferred in the starting XI to Pepe Reina, was kept busy as he had to push away Wayne Routledge's dangerous cross before watching two well-struck shots by Wilfried Bony go narrowly wide.
Bony had a glorious chance to put his side ahead in the 22nd minute as Pablo Hernandez's delightful pass split the Napoli defence open, but Rafael was swiftly off his line to produce an excellent, but unorthodox, sprawling save as the ball struck him in the face.
Monk's men continued to press forward, with Ben Davies' dangerous cross fisted away by the busy Rafael before a weak attempted clearance presented possession to Jose Canas on the edge of the box but he could only volley over the crossbar.
Six minutes from the interval another chance went begging, a short corner routine catching Napoli napping with Hernandez's cross to the back post headed over the bar by Ashley Williams, with the centre-back left holding his head in his hands.
There was still time for Rafael to be called into action again before the break, reaching to get his finger-tips to a rising Routledge shot and then pushing Angel Rangel's header from the resulting corner over the top.
Rafael's game was brought to a premature conclusion at half-time by injury, with Reina sent on in his place, and the former Liverpool man had a quieter start than his team-mate, with Hernandez's quick feet taking him into the box only for Henrique to block his 55th minute effort.
Williams was called upon to produce an excellent tackle to deny Hamsik after the Napoli man had combined well with Jose Callejon to get in behind the Swansea defence on the right of the box.
The duo combined again in the 63rd minute to create another opportunity, but Callejon could only toe-poke his shot wide after neatly nut-megging Davies on the edge of the area.
Williams had his head in his hands again in the 67th minute after meeting Jonathan de Guzman's excellent free-kick to the far post with a firm header which was parried away by Reina.
Another opportunity was carved out by Swansea in the 76th minute, but again Reina was equal to Routledge's curling shot as he dived to his left to fist away, with the chance created by the returning Jonjo Shelvey after stepping off the bench as the game ended goalless.