Leicester travelled to Naples knowing they would guarantee a top-two finish in Group C if they avoided defeat, but first-half strikes from Adam Ounas and Elmas put the hosts ahead on Thursday.
Jonny Evans and Kiernan Dewsbury-Hall quickly restored parity in a chaotic opening 45 minutes, only for Elmas to net again after the break and decisively edge Napoli back in front.
Luciano Spalletti's side settled for second - with Spartak Moscow defeating Legia Warsaw, following a 98th-minute penalty save, to top the group - and go into a two-legged play-off in February, while the third-placed visitors face a tie at the same stage of the Europa Conference League.
Brendan Rodgers' men began the night top of Group C, but much of the pre-match talk was of Leicester having to leave behind seven players through illness.
Rodgers confirmed some of the unnamed players had tested positive for coronavirus while others had been left at home as a precaution.
But it was a strong Foxes team that took to the field, which included Belgium midfielder Youri Tielemans returning from a five-match absence with a calf problem.
Bottom-club Antwerp upset Olympiacos 1-0 but the Greek club still progressed with a second-place finish in group D, while Real Sociedad beat PSV 3-0 to snatch second in Group B behind Monaco, who drew 1-1 with Sturm Graz.
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