The 29-year-old, who started at right-back in Saturday's LaLiga clash, was hurt by a challenge with Sergio Busquets and was substituted with two minutes left of the first half.
The injury is almost certain to keep Vazquez out of action for the rest of 2020-21.
The loss of Vazquez, who has made 34 appearances in all competitions this term, is a blow to Zinedine Zidane's hopes of sustaining a LaLiga and Champions League challenge.
Dani Carvajal remains sidelined with a hamstring problem after playing only twice since the turn of the year, leaving Alvaro Odriozola as Madrid's remaining option at right-back.
Los Blancos went top of the table with their Clasico win, although Atletico Madrid will return to the summit if they avoid defeat to Real Betis.
They face Liverpool at Anfield on Wednesday in the second leg of their Champions League quarter-final tie, leading 3-1 from the first leg.