Federico Valverde and Vinícius Júnior scored for the defending champions on Saturday, four days after they lost at Lille 1-0 in the Champions League to halt a 36-game unbeaten streak in all competitions.
Beating third-placed Villarreal put Madrid level on 21 points with leaders Barcelona, who visit Alaves on Sunday.
Carvajal was hurt in second-half stoppage time at Santiago Bernabeu Stadium, going down in pain after kicking the leg of an opponent. The Spain international was crying when he was carried off the field on a stretcher.
Real Madrid's Dani Carvajal grimaces in pain after being injured against Villareal. (AP PHOTO)
"It looks like it's a very serious knee injury," Madrid coach Carlo Ancelotti said. "Everyone is sad and worried. It's something that happens very often because of the calendar and it has happened to a very important player in out squad."
Villarreal, who were coming off two straight wins, had only one attempt on target. Madrid had two, scoring on both of them.
Valverde scored in the 14th minute with a long-range shot that deflected in off a defender. Vinícius struck from outside the area in the 72nd into the top corner.
Celta Vigo striker Iago Aspas was sent off in a 1-0 win at last-placed Las Palmas after being issued consecutive yellow cards for protesting.
He complained about the referee's decision to show a red card to teammate Ilaix Moriba and received the first yellow in the 54th minute. Aspas' second came in the 56th.
Celta held on despite playing two men down to the end of the match. Borja Iglesia scored a 28th-minute winner in the visitor's only attempt on goal.
The result ended a three-game winless streak for Celta.
Las Palmas, the only team yet to win in the league, haven't won in 23 straight league matches going back to last season, when they barely avoided relegation.
Jorge de Frutos scored a pair of second-half goals for Rayo Vallecano to win at Valladolid 2-1, while Espanyol defeated Mallorca 2-1 at home and Osasuna drew at Getafe 1-1.
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