Nacho Monreal drew an early penalty and Hazard, whose season has been plagued with injury thus far, stepped up to dispatch the spot kick and give Real an early lead in this all-important group-stage match.
The goal was Hazard's first in the Champions League since scoring from the penalty spot for Chelsea against Qarabag on Nov. 22, 2017.
In a bizzare sequence of events just after the half-hour mark, Arturo Vidal was shown back-to-back yellow cards for dissent after arguing with the referee for a penalty kick, leaving Inter to finish the match with only 10 players.
Rodrygo came off the bench just before the hour mark and made an instant impact, meeting a Lucas Vazquez cross at the back post and smashing a blistering first-time volley past Samir Handanovic to double Madrid's advantage.
Inter had another penalty waved away when Roberto Gagliardini went down early in the second half before Real struck again.
Ten-man Inter couldn't find a way back into the match as Zinedine Zidane's side picked up a big win away from home.
"Win, lose or draw, the important thing is to play well. Today we deserved to win and we had the match under control until the end," said Zidane, whose side were held 1-1 at Villarreal in La Liga on Saturday.
"It doesn't change anything. We also played well the other day and deserved three points."
The result puts Real Madrid in second place on seven points after four games, one behind leaders Borussia Monchengladbach, while Inter are stuck at the bottom of the table on two points.