The deal, announced in April, was signed at a ceremony in Tehran and the first 30 planes are due for delivery between 2022 and 2024.
"I'm glad that we can... Upgrade the air fleet in an appropriate manner so they can take over regional markets," said Labour Minister Ali Rabii at the signing, according to the ISNA news agency.
However, the deal could be scuppered if US President Donald Trump goes ahead with threats to reimpose sanctions on Iran.
Washington must renew sanctions waivers every few months to keep the deal alive, and another set of renewals is due in the coming days.
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Even if Trump sticks by the nuclear deal, the Boeing sale must be cleared by the US Office of Foreign Assets (OFAC).
"We have finalised the deal and now we are waiting for OFAC permission within the next month," Aseman spokesman Amir Reza Mostafavi told AFP.
Boeing, which is also building 80 planes for national carrier Iran Air, faces heavy criticism from US lawmakers who say Iranian airlines have been used to ship weapons and troops to Syria and other conflict zones.
The aerospace giant has therefore emphasised the employment potential of the deals, saying in April that the Aseman contract "creates or sustains approximately 18,000 jobs in the United States".
Iran has been desperate to renew its ageing fleet of planes, but was largely blocked from dealing with major aircraft manufacturers until the 2015 nuclear accord.
OFAC approved the sale of the 80 Boeing as well as 100 Airbus planes to Iran Air. The first few Airbus jets have already arrived in Tehran.
Aseman currently has a fleet of 36 planes -- half of them the 105-seat Dutch Fokker 100s.
Its three Boeing 727-200s are almost as old as the Islamic revolution, having made their first flight in 1980.