The game's first real chance fell to Atalanta and it took a great piece of defending from Fredy Guarin to clear off the line, after Samir Handanovic presented an easy chance to Michele Canini in the 10th minute.
Inter made the home side pay minutes later as Ricky Alvarez brought down a header before firing past Consigli's near post from close range.
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However, Atalanta hit back with an equaliser in the 25th minute following great forward play from German Denis, who moved smartly to get a yard on the defender and head adroitly past a helpless Handanovic to bring the score to 1-1 going into half time.
Inter raced out of the traps after the break as Esteban Cambiasso let fly from range, but Consigli did well to turn the ball away for a corner.
Consigli was called into action again in the 63rd minute as Mateo Kovacic exchanged passes with Alvarez, before being denied by Atalanta goalkeeper who used his feet to keep out the effort.
Inter thought their hard work had paid off when Mauro Icardi produced some great skill before crashing an effort towards goal, but his shot came back off the post despite beating Consigli as the away side continued to hunt for the lead.
As Inter pushed for three points, Atalanta were presented with a great opportunity to win the game in the final stages as Mario Yepes found himself in acres of space, but substitute keeper Juan Carrizo managed to pull off a fine save from six yards to deny the defender.