The Gunners only needed a draw to top the pool and looked on course to gain that result, only to be undone by former transfer target Gonzalo Higuain in the 73rd minute.
Three minutes later, Arteta was shown a second yellow card for a late challenge on Jose Callejon, who grabbed a second goal in time added on.
However, it was not enough for Rafael Benitez's side as a late Kevin Grosskreutz goal gave Borussia Dortmund a 2-1 victory in Marseille in the other game in Group F. Three teams were level on 12 points, but the Germans finished top and Arsenal claimed second spot due to the three-way head-to-head.
Arsene Wenger's men must now face one of five top seeds in the last 16 - Barcelona, Bayern Munich, Real Madrid, Atletico Madrid or Paris Saint-Germain.
Wenger made two changes following the 1-1 draw at home to Everton in the Premier League on Sunday. Tomas Rosicky and Mathieu Flamini came in for the benched Jack Wilshere and Aaron Ramsey.
Napoli were without goalkeeper Pepe Reina and star man Marek Hamsik due to injury. Pablo Armero, Valon Behrami and Dries Mertens were drafted in after a 3-3 Serie A draw with Udinese on Saturday, with Lorenzo Insigne and Gokhan Inler among the substitutes and Anthony Reveillere missing out.
Early on, Goran Pandev got away down the left and his angled shot was held by Wojciech Szczesny.
It then needed a solid block from Germany centre-half Per Mertesacker after left-back Armero had broken quickly on the counter-attack.
Arsenal's best chance of the first half came just after the midway point. Rafael Cabral was forced into an impressive save to deny Olivier Giroud, who was played into space on the left side of the area after Flamini's pass and unleashed a fierce shot. The ball fell at the feet of Mesut Ozil but he slipped in the process of shooting.
There was a let-off for the Gunners on 37 minutes when full-back Christian Maggio got clear down the right and Szczesny came racing out to the edge of the penalty area. The chip from the Napoli captain, though, floated over the crossbar.
The Poland keeper was then almost embarrassed when his clearance bounced off the head of Higuain and just wide.
Arsenal were more positive at the start of the second half as they looked to close in on the positive result needed to top the group, with Ozil firing wide from 20 yards after being set up by Santi Cazorla.
Napoli knew they had to up their game and left-back Armero blasted straight at the Arsenal keeper when played into the penalty area on 57 minutes.
Callejon then dragged a shot across goal and wide from the right - much to the disapproval of the home support.
On 73 minutes, Armero's ball in from the left was touched back to Callejon by Higuain, and then quickly returned to the Argentinian. He spun on the edge of the area and sent a low shot past Szczesny for an excellent finish.
Arteta received his marching orders soon afterwards, but the Gunners stayed strong until stoppage time when Callejon notched another goal from an assist by substitute Insigne - but the final whistle blew almost immediately afterwards, sending Napoli into the Europa League round of 32.
Arsenal face another tough away game at lunchtime on Saturday when they take on Premier League title favourites Manchester City at the Etihad Stadium.
The Gunners will then discover their Champions League last-16 opponents when the draw is made on Monday, but are now assured of a difficult two-legged tie after slipping up in Italy.