The company will build Al Bayt stadium in al-Khor city, north of Doha, shaped after the Bayt al-Shaar black and white traditional bedouin tent.
The Italian firm said on its website it had beaten competitors from France, Austria and Asia for the contract to build a stadium that could accommodate 70,000 spectators.
The contract includes €716 million euro for construction and over €53 million euro for operations and maintenance, it said.
In addition, the company won a contract worth €300 million to build main urban infrastructures in Shamal, some 100 kilometres (186 miles) from the capital.
Salini Impregilo is already working on building part of Doha Metro and a water project.
Energy-rich Qatar is undergoing a huge spending splurge on infrastructure, worth around $200 billion, much of it related to the World Cup.
Qatar's successful bid was originally to host the tournament in the months of June and July but a Fifa decision earlier this year decided on an unprecedented switch to play the tournament in the months of November and December 2022 because of the extreme summer temperatures in the Gulf.