Irvine received a second yellow card after 38 minutes with the score 1-1 and Burton then conceded three times to slip to a second league defeat.
"I've got no problems with the first yellow card but the second one was harsh," said Clough.
Burton will try to make a loan signing before Tuesday's trip to Middlesbrough.
Clough's side will head to Teesside with no points from their opening two league matches and will be without suspended Irvine at the Riverside Stadium.
"It was a trip on the edge of the penalty area," said Clough. "Luke Murphy got brought down in the same way and I asked the fourth official what the difference between the two challenges was and he couldn't tell us.
"The head of referees keeps using this analogy which I never agree with about 'your players make mistakes as well', but our players have got 11 players against them - the officials don't have anybody.
"The referee and both linesmen made incorrect decisions which have had a big effect on the game."
Burton Albion manager Nigel Clough labelled the sending off of Jackson Irvine during his side's 4-1 defeat at Hull on Saturday as "harsh".