They accused Perry of using the 2008 track – which features Lecrae – without permission, tarnishing its "devoutly religious message" with images of "witchcraft" and "black magic."
It comes from Flame's Grammy-nominated album Our World: Redeemed.
Dark Horse topped the US Billboard chart for several weeks in January.
The group of rappers – Flame aka Marcus Gray, Lecrae Moore, Emanuel Lambert and Chike Ojukwu – are seeking damages from Perry, guest artist Juicy J, Capitol Records and songwriters Dr Luke and Max Martin for copyright infringement.
Court papers in St Louis, Missouri, show they also want an injunction and profits gained from the unauthorized use of their track, stating Perry had performed Dark Horse at sold-out concerts throughout the US and sold millions of copies.
"The devoutly religious message of Joyful Noise has been irreparably tarnished by its association with the witchcraft, paganism, black magic, and Illuminati imagery evoked by the same music in Dark Horse," said the rappers' complaint.
They continued: "Indeed, the music video of Dark Horse generated widespread accusations of blasphemy and an online petition signed by more than 60,000 demanding removal of an offensive religious image from the video."
The video for Dark Horse was edited in February this year following the complaints, with a pendant featuring the Arabic word for God being digitally "scrubbed" out of the scene.
More than 440 million people have watched the video on YouTube since February this year.
Flame said the track's similarities to Joyful Noise were brought to his attention by his DJ, Cho'zyn Boy. The DJ compiled an audio comparison of the two songs, published by Rapzilla, after noticing comments about them on Twitter.
They claim Perry and her collaborators "never sought or obtained permission" to use Joyful Noise for "creating, reproducing, recording, distributing, selling, or publicly performing" Dark Horse.
Representatives for Perry, who released a Christian Rock album under her birth name Kate Hudson in 2001 before she was famous, said she was unable to comment because of the legal proceedings.