Hurley posted a fuzzy 2½ minute long video on his new website MixBit, that showed West proposing to Kardashian while an orchestra played.
The video also showed the ensuing celebration with Kardashian's family and invited guests toasting the newly-engaged couple.
Kardashian and West are seeking unspecified damages from Hurley and his company, Avos Systems Inc.
"Exclusive rights, such as those sold by plaintiffs to publication of video of the event, are particularly valuable," the lawsuit states. "If people violate these rights ... they are of substantially diminished value."
A message sent through MixBit seeking comment from Hurley was not immediately returned.
West proposed to Kardashian on her 33rd birthday; the couple's first child North West was born in June.
The lawsuit was first reported by celebrity website TMZ.