Milanov, capped 23 times by Bulgarian, was injured in the closing minutes of the first half and was rushed to hospital.
The 28-year-old lost two teeth when Portuguese Cadu, sent off for the incident, mistimed a challenge for a loose ball and another three teeth were removed in hospital.
"It's a crime," Vaslui coach Viorel Hizo told local media. "It's not the first time Cadu has committed such a foul and I'm curious what his punishment will be."
Vaslui are fourth in the standings while titleholders Cluj, who finished third in their Champions League group behind Manchester United and Galatasaray this season, are sixth.