The men were recruited in Nepal to work in a hotel in Jordan, but were later told they would have to work at a US air base in Iraq, their lawyers said.
Twelve of the men were kidnapped and killed by Islamic militants while being transported inside Iraq.
The 13th man was made to work against his will at the air base, lawyers said.
The execution-style killing of the hostages was recorded by the extremists and posted on a website.
The incident sparked riots in Nepal with angry demonstrators targeting a mosque, some government buildings and offices of employment agencies.
At least two people were killed in the protests.
The lawsuit filed in the US District Court in Los Angeles on Wednesday alleged “that the illicit trafficking scheme… was engineering by KBR and its subcontractor”, identified as Daoud & Partners.
The men, between the ages of 18 and 27, were recruited “to work as kitchen staff in hotels and restaurants in Amman, Jordan”, said a statement from Cohen, Milstein, Hausfeld & Toll, one of the law firms handling the case.