The visitors created the first chance of the match when Marten de Roon burst to the edge of the box before firing an effort wide of the target.
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On the stroke of half time it was Atalanta who again threatened to make the breakthrough when German Denis rose to meet Kristian Raimondi's cross before heading narrowly over the top corner.
Chievo came out strongly in the second half and Massimo Gobbi had an early chance with a strike from inside the box that went a few yards wide of the post.
Paloschi then stepped up to take the resulting spot kick, but his effort was kept out by Marco Sportiello.
The keeper was soon called into action again to turn wide Ivan Radovanovic's low free kick that was heading into the bottom corner.
However, Chievo soon made their numerical advantage count as Birsa was found inside the box by Riccardo Meggiorini, before firing in a low drive at the near post.
Atalanta saw their misery compounded in stoppage time as Jasmin Kurtic picked up two quick bookings and was sent off, as they ended with nine men for the second successive away league match.