The Uruguayan went down after a second-half tackle by goalscorer Samuel Eto'o but referee Howard Webb waved away his appeals for a spot-kick - moments after the striker had also fallen to the floor following a penalty box challenge by John Terry.
"I think there is another big case for a penalty on Hazard," he said. "Something is happening, week after week, I don't know why referees don't see it. He's a fair boy and I don't know why this is happening.
"On the other situation with Suarez I would resolve this differently with a yellow card for Luis Suarez. I keep saying I hate players trying to provoke situations. Luis is an amazing player and I have lots of respect for his quality and his attitude of trying to win every game but sometimes he tries too much.
"Eto'o just put his body in front of him. It looks like somebody shot him."
"He does everything to win and Brendan (Rodgers, the Liverpool manager) is doing a fantastic job on him because he's changed. There's no doubt that he's changed.
"But when the situation is like this and you are losing comes the nature of the player, the wild nature of the player or the cultural nature of the player.
"Culturally people from that area (South America), they like it. It's not also that area. There's a corner in Europe where I belong to, that they also like (diving)."
Mourinho added in his press conference: "We don't like people with simulation. This situation in the box, Azpilicueta has the ball, Azpilicueta is leaving the box with the ball and now he's (Suarez) doing an acrobatic swimming pool jump."
Although Eto'o scored the winner, he was perhaps lucky to escape punishment for an early challenge on Jordan Henderson. It was from the resulting free-kick that Liverpool scored through Martin Skrtel.
However, Mourinho sought to take the blame, saying he had told Eto'o to take a more physical approach.
"I didn't see it, I don't know which situation you are referring to," he said when asked about the challenge on Henderson.
He then added: "For some weeks I have been criticising him a lot internally for his failure to adapt to what the Premier League is in terms of physicality so if he had a tackle deserving of a card it was my fault because I'm asking him to adapt to this."
In responding to Mourinho, Liverpool boss Rodgers pointed out Eto'o was fortunate to still be on the pitch.
"He (Eto'o) should've been sent off," Rodgers said.
He then discussed Mourinho's comments on Suarez and added: "Luis will always provoke a challenge from defenders, that's why he's world class.
"What he probably doesn't expect is to provoke a challenge from someone who's not involved in the contact.
"He's running to challenge (Cesar) Azpilicueta and Eto'o is street wise, he's cute, he blocks him. On another day you could give that as a penalty, because it's obstruction in the area.
"Of course, he (Mourinho) will defend his player and I will defend mine."
Mourinho began his post-match comments by paying tribute to the efforts of Premier League players around the country for their performances throughout the busy festive period.
"Premier League players in this country deserve fantastic respect because what they do is amazing," he said. "To play in this period is beautiful. I love it. I keep saying I love it. The players are monsters in a good sense of the word.
"It is fantastic propaganda for what football is."
At the centre of most of what was good about Chelsea was Hazard and team-mate David Luiz hailed the Belgian for his display.
"Of course it is not very good to (concede) a goal in the first minutes but there was a really strong mentality in the team, we gave everything on the pitch and it was magic tonight," Luiz said on Sky Sports 1.
"Eden had a great game tonight and I'm happy we have players like Eden in the team because then I see amazing football."
The win keeps Chelsea in third, just two points from leaders Arsenal. "We will keep fighting for the title," Luiz added.