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.
"To lose your first two home games of the season isn't pleasant," said an uncomfortable Allegri who declined to comment on his relationship with club vice-president Adriano Galliani.
Earlier, Cagliari's Marco Sau came off the bench to snatch a late headed goal in a fortunate 1-1 draw at Palermo.
First the keeper 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.
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 they have been deducted, and lifts them out of the relegation zone.
Allegri is hoping the start of Milan's Champions League campaign can kick-start their stuttering season.
"I believe that we have to continue to work, but fortunately on Tuesday we have our match in the Champions League that will give us the opportunity to redeem ourselves and get going again" he told reporters.
"At the moment we're playing a bit in bursts, so we should look to play with a bit more conviction, otherwise it all becomes more difficult," added Allegri.
"On Tuesday we have to play a different game, not technically, but we have to have more conviction and believe in ourselves."
