Cattermole was sent off for the seventh time in his Premier League career after a studs-up challenge on former team-mate Ahmed Elmohamady.
But Poyet looked to defend the midfielder after he became the first of two Sunderland players to receive straight red cards, Andrea Dossena also heading down the tunnel moments after Cattermole was dismissed.
"There were too many decisions to make for the referee," said Poyet. "I'm convinced (Cattermole) didn't deserve a red.
"Sometimes people just see Lee Cattermole making a tackle and it's not allowed. It should be a red card or nothing but we need to be fair to him and this is not fair because this is a game where people have to make tackles.
"This is Lee Cattermole but people put things in the mind of other people. If you keep talking about it then people will listen."
Poyet made no attempt to defend the dismissal of Dossena with his opposite number, Steve Bruce, insisting the Sunderland full-back "deserved three red cards" for his shin-high challenge on David Meyler.
However, the Black Cats boss did feel aggrieved that referee Andre Marriner failed to take action against Tigers defender Paul McShane who clattered into keeper Kieren Westwood - a challenge which eventually forced Westwood off with injury.
"The Kieren Westwood tackle nearly took his head off," said Poyet. "We talk about being fair and professional - but at the end of the day we lost a player that we shouldn't have."
Westwood's injury led to the introduction of former Hull loanee Vito Mannone and the substitute keeper performed well as the nine men kept the Tigers at pay for the entire second half. Sunderland could even have pinched a point only for substitute Adam Johnson to be denied by Steve Harper late on.
"It was difficult to compare the second 45 minutes with the rest of season but tactically it was our best 45 minutes of the season," said Poyet.
"It was outstanding. Tactically you cannot be better and to have a couple of chances, especially the Adam Johnson one, I'm very happy. On another day that goes in and we would be celebrating a point. He's in the dressing room devastated he did not score but he did a terrific job for us. He was a team player for us today."