Third in Serie A last season, Napoli were too strong for Nice, winning 2-0 in France with Callejon and Lorenzo Insigne both scoring as they progressed 4-0 on aggregate.
Beaten by eventual winners Real Madrid in the last 16 last season, Maurizio Sarri's side took the lead when Marek Hamsik deposited a low cross from the left for Callejon to convert at the back post in the 48th minute.
"Napoli must keep on playing in the Champions League every season, because this is where we belong. We are happy with the qualification," said Insigne.
Nice, for whom recent signing Wesley Sneijder started, were thoroughly outplayed over the two legs and will have to settle for a place in the group stage of the Europa League instead.
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They went behind on the night to an Eljero Elia goal but Sergio Escudero headed in a Jesus Navas cross early in the second half for the home side.
Great play from Nolito then set up Wissam Ben Yedder to seemingly end any doubt, but Edin Visca levelled the scores again on the night and an Emre Belozoglu free-kick struck the woodwork in the last minute as Sevilla came within a whisker of going out on away goals.
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Celtic went through 8-4 on aggregate despite losing 4-3 to Astana in the second leg of their play-off in Kazakhstan as a second-half wobble gave their hosts hope of pulling off a miraculous recovery.
Brendan Rodgers' side had won 5-0 in the first leg in Glasgow and a super Scott Sinclair strike in the 34th minute should have killed the tie.
Astana, who had gone ahead on the night through a Kristoffer Ajer own goal, were given something to cling onto as a goal by Serikzhan Muzhikov and a Patrick Twumasi brace made it 4-1 on the night.
"I have mixed emotions because we didn't defend well enough," Rodgers told BT Sport after a scare that brought back memories of the same stage 12 months ago.
Celtic beat Hapoel Beer-Sheva of Israel 5-2 at home in the first leg of the play-offs last year and then wobbled in the return, losing 2-0 to scrape through.
"We had our Beer-Sheva moment. I thought we were going to get through this campaign without the anxiety. The players were on the ropes for 15 or 20 minutes but then they regained composure."
Maribor of Slovenia also progressed, beating Hapoel Beer-Sheva 1-0 with a Mitja Viler strike as they won the tie on away goals.
The remaining five ties will conclude on Wednesday, with the highlight seeing Liverpool take a 2-1 lead into their play-off second leg against Hoffenheim at Anfield.
The draw for the lucrative group stage will be held in Monaco on Thursday.
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