Rafael Benitez's men were made to work for the win but clinched their place in the next round thanks to the Italian's second-half strike
The Partenopei went into the game as favourites, with the visitors just five points above the relegation zone in Serie A, and Rafa Benitez's men ensured there was no cup shock at San Paolo.
However, the home side got off to the worst possible start, as Atalanta took the lead on 14 minutes when Giuseppe De Luca side-footed home from close range at the end of a counterattack.
But their lead was to last just 60 seconds. A left-wing cross from Anthony Reveillere found Callejon at the back post and the former Real Madrid man struck a wonderful volley back across goal and into the far corner.
The game remained level at the half-time interval and Napoli had to wait until the final 20 minutes before taking the lead for the first time on the night.
Lorenzo Insigne was the man who got the goal, with the Italy international firing home on 72 minutes after Gonzalo Higuain's floated pass was deflected into his path by Cristiano Del Grosso, as the defender appealed for offside.
Atalanta players surrounded officials to register their protests over the decision of the linesman not to raise his flag for offside, but the goal stood, and eight minutes later Benitez's men made absolutely sure of the win.
Gokhan Inler played in Callejon with a ball over the top of the back four and the Spaniard lashed the ball home to complete a 3-1 win and send Napoli into the quarter-finals of the Coppa Italia.