The setback piled the pressure on Milan manager Massimiliano Allegri whose team are stuck in mid-table on three points having lost two of their first three league games this season.
Atalanta, who had two points deducted for their role in the latest match-fixing scandal, never let Milan relax with German Denis causing problems with his hold-up play and movement.
The Argentine, nicknamed The Tank because of his strength, could have opened the scoring after 20 minutes when he pounced on a Francesco Acerbi error but his shot hit the post.
Six minutes later Milan could have opened the scoring themselves after forcing Atalanta's Andrea Consigli to make a spectacular double save.
First he kept out Stephan El Shaarawy's long-range strike and then the rebound off Gianpaolo Bellini to prevent a comical own goal after the defender had raced back to clear.
The second half was short on chances with both sides seeming content to shoot from distance but Atalanta struck after 64 minutes when Cigarini raced onto a Denis lay-off and skidded a low shot across Christian Abbiati's body from 25 metres.
Milan reacted by being more aggressive and pushing their opponents back and would have equalised but for Giampaolo Pazzini and Antonio Nocerino failing to make contact with good crosses with the goal at their mercy.
However, Atalanta could have doubled their lead with six minutes left but Cristian Raimondi wasted the chance when he finished a smart counter-attack by shooting straight at Abbiati.
Despite Milan's continued pressure, Stefano Colantuono's Atalanta side held on for a victory that puts them on two points, taking into account the two points that have been deducted, and lifts them out of the relegation zone.
Earlier, Cagliari's Marco Sau came off the bench to snatch a late headed goal in a fortunate 1-1 draw at Palermo.
Palermo took the lead in the 41st minute through Uruguay midfielder Arevalo Rios who arrived on the edge of the area to open his account for the season by ramming home a deflected shot from a Fabrizio Miccoli cut back through a crowd of players.
Massimo Donati thought he had doubled the Sicilian club's lead when he headed home Miccoli's free-kick with eight minutes to go but it was controversially ruled out for offside.
Cagliari took advantage of their luck six minutes later when striker Sau headed home the rebound from Samir Ujkani's save.
The draw leaves the Sardinians in 15th place with two points ahead of Palermo who earned their first point of the season.