Fredy Guarin had an early sight of goal for the hosts when he was picked out on the edge of the box, but he could only place his shot wide of the post.
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The midfielder came close again moments later as he worked himself some space to shoot before firing a long-range effort narrowly over the bar.
Atalanta continued to threaten and were inches away from taking the lead when Denis met Bonaventura's cross and powered a header against the bar.
In the 35th minute Atalanta took the lead through Bonaventura, who was picked on the edge of the box and powered a low strike into the bottom corner.
Their lead didn't last long as within a minute Inter were level, as Icardi burst into the box and drilled an effort past the helpless Consigli.
Danilo D`Ambrosio had a chance to put his side ahead in the final minute of the half, but after being picked out inside the box he could only volley an effort wide of the post.
After the break Guarin came close to scoring when he received the ball in space outside the box and fired a strike against the bar.
Franco Brienza came close to giving the visitors the lead when his attempt was cleared off the line, before Jonathan hit the crossbar for Inter from close range.
With a minute of the match left to play, Atalanta stole all three points as Bonaventura arrived at the back post to head Brienza's free kick past Handanovic and into the net.