Villarreal must overturn a one-goal deficit in Monaco if they are to reach the group stages having fallen to a 2-1 defeat at home.
It took just three minutes for 2004 runners-up Monaco to lead through a penalty from rumoured Manchester United target Fabinho.
Alexandre Pato, a loanee with Chelsea last term, equalised for the Yellow Submarines.
But Bernardo Silva claimed a winner for Monaco, who had Benjamin Mendy sent off.