Carlos Carmona fired the home side in front in the 21st minute, when his strike from just outside the box took a huge deflection and left Michael Agazzi with no chance as it found the back of the net.
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After the break Giacomo Bonaventura came close to doubling his side's lead, but Consgli kept out his powerful shot.
In the 68th minute Chievo were reduced to 10 men, as Bostjan Cesar picked up his second yellow card of the afternoon.
The red card seemed to galvanise the visitors and they found an equaliser four minutes later, when Dario Dainelli met Tiberio Guarente's free kick to power a header past Consigli and into the bottom corner.
With five minutes left to play Atalanta scored their second, as German Denis laid the ball off for Cigarini to strike on the edge of the box, and the midfielder made no mistake to bury a low drive and earn his side the three points.