The Welsh side looked set to make it 2-2 with two minutes remaining in the Stadio San Paolo last night, only for the on-loan Liverpool goalkeeper to repel Dwight Tiendalli's header.
Swansea, playing in a knockout round in European competition for the first time, turned in an impressive display.
They almost opened the scoring in the 11th minute when Pablo Hernandez fed Marvin Emnes on the right of the box and his goalbound shot beat Reina but was cleared off the line by Raul Albiol.
Lorenzo Insigne put the hosts ahead five minutes later when he beat a haphazard offside trap before lobbing the Swansea goalkeeper Michel Vorm.
Swansea were level on the half-hour when Wilfried Bony cleverly sent Jonathan De Guzman through and the Dutchman surged forward before drilling an 18-yard shot past an onrushing Reina.
However, Napoli regained the lead after 78 minutes when Ben Davies failed to clear the ball in the box and Gonzalo Higuain struck the ball past Vorm. Gokhan Inler added a third for the Italian club in injury time.