The Public Enemy rapper got into a verbal argument with the woman and the 17-year-old before it turned violent, according to The AP.
The hip-hop star, born William Jonathan Drayton Jr, apparently cut his finger with one of the knives, while the woman claims she had an earring ripped out during the alleged incident.
A Las Vegas police statement said of the fiancée's 911 message: "The caller advised she had been involved in a physical altercation with her fiancé, who was identified as William Drayton."
The 53-year-old was taken in by police on suspicion of assault with a deadly weapon and battery.
He is currently being held at the Clark County jail on $23,000 (£14,300) bail, pending an initial court appearance on Thursday.
This summer, Flavor Flav avoided a prison sentence after paying towards over $100,000 he owed in back child support.