The hosts extended their unbeaten run to seven games but a late goal from Josip Ilicic ensured the spoils were shared.
The home side regained the lead through an Arkadiusz Milik strike at the start of the second half and looked to be heading for all three points before Ilicic's 86th-minute equaliser.
The Napoli players and coaches were incensed by the referee's decision to allow the goal, and manager Ancelotti was shown a red card moments later for remonstrating about a penalty decision which did not go his team's way.
Napoli did not have long to wait for the opening goal, however, as Callejon turned provider with a pinpoint cross for Maksimovic to head home.
The home side came close to doubling their lead less than 10 minutes later when Gollini denied Fabian Ruiz, with Milik heading the loose ball wide.
The visitors levelled through Swiss midfielder Freuler, who took a touch after being played through by Rafael Toloi and slotted the ball past Meret.
The Partenopei continued to put pressure on the Atalanta defence throughout the second half and were rewarded in the 71st minute.
Milik rounded the goalkeeper before tapping into an empty net.
But, for the second time in the match, Napoli let their lead slip as Ilicic beat the offside trap before firing home.
The home side protested vehemently, with Lorenzo Insigne shown a yellow card and Ancelotti sent to the stand because of his protests over a penalty that was not awarded.
Ilicic almost snatched a winner midway through eight minutes of stoppage time but Meret came to the rescue.