The group have been accused of wrongfully sampling material from a hip-hop label's catalogue on their first two albums Licensed to Ill and Paul's Boutique.
TufAmerica allege that the band and Universal Music Group used a song by the group Trouble Funk titled 'Say What', and that it can be recognised in Beastie Boys track 'Shadrach'.
They also claim that 'Car Thief' samples Trouble Funk's 'Drop the Bomb', and allege that more material of the same track also shows up on Beastie Boys songs 'Hold It, Now Hit It' and 'The New Style' from their 1986 debut Licensed to Ill.
The surviving two members of Beastie Boys - Mike 'Mike D' Diamond and Adam 'Ad-Rock' Horovitz have not commented on the lawsuit, but the prosecuting company's attorney has released a statement following Yauch's death on May 4 at the age of 47.
Kelly Talcott told E! News: "I was very sorry to hear of Adam Yauch's untimely passing, and can assure you that the unfortunate timing of the filing of TufAmerica's complaint had nothing to do with his health.
"On behalf of myself and TufAmerica, I offer our condolences to Adam's family, friends, and fans."
Listen to Trouble Funk's 'Drop the Bomb' below: