The Belgian winger lasted less than 30 minutes before limping off the field as Zinedine Zidane's side would eventually fall 2-1 to the visitors.
After the game, Zidane expressed his hope that Hazard's injury wouldn't keep him out long.
"I think and hope it's just a knock," he said at his post-game press conference. "I hope it's not muscular."
Hazard was making just his third appearance in La Liga this season as the 29-year-old's campaign has been heavily disrupted by injuries, having missed the start of the campaign due to a muscle issue.
He played for the first time in 2020-21 when he came off the bench in the Champions League draw with Borussia Monchengladbach in late October and returned to the starting XI four days later in the league match against Huesca, scoring the first goal as they ran out 4-1 winners.
A few days later, however, he tested positive for Covid-19 and had to be sidelined for a short period.
Hazard was in action again last week when he started in the 1-1 draw against Villarreal in the Spanish top flight.
He then featured in the 2-0 win at Inter in the Champions League in midweek, in which he opened the scoring after just seven minutes.
Madrid fell behind against Alaves just five minutes into Saturday's clash when ex-Arsenal and West Ham attacker Lucas Perez converted a penalty.
Hazard had a chance to level the score later in the half when he collected the ball near the halfway line and drove towards goal. The former Chelsea star's effort was denied by Alaves goalkeeper Pacheco, however.
Madrid would go down two goals early in the second half when a wayward Thibaut Courtois pass was collected by Joselu 25 yards out and deposited into an empty goal. Casemiro grabbed a late consolation but ultimately Alaves held on for a famous win.
Hazard has made 28 appearances in all competitions for Madrid since he joined from Chelsea in the summer of 2019.