Herrera bodychecked Hazard to earn his first caution before a late trip on the same player left United with a numerical deficit in just the 35th minute.
N'Golo Kante lashed home a brilliant goal after the break as Chelsea pressed home their advantage and left Jose Mourinho to lament another sorry return to his former club.
Mourinho was predictably frustrated by calls that he perceived went against his team, but Conte hinted that he agreed with Michael Oliver's decision.
"I think that Hazard started the game and he couldn't play football," Conte told the BBC. "All the people can look at this and then judge the situation."
Conte added in his news conference: "When you play against players with great talent, sometimes you try to intimidate them and referees must protect these players because they could go out with a bad injury."
Runaway Premier League leaders Chelsea showed their mettle, dominating a United side that has reacted well to a 4-0 demolition at Stamford Bridge earlier in the season.
With Tottenham awaiting in a Wembley semi-final, Conte hailed the big-game instincts of his players.
"It was a good performance against a team that is very strong, with good players," he said.
"For me, United are the best squad in the league. They have a lot of good players and we must be pleased to go in the next round and to beat a great opponent.
"In a semi-final it's normal for the draw to be difficult, but also in this quarter-final it was the same, very difficult against United.
"But now we go to Wembley and we are ready to play against Tottenham, against a really strong team but now we must be satisfied for the result and for what we are doing in this season."