A mix up between Frederico Venancio and Joe Wildsmith saw the Owls go behind after just eight minutes through David Davis and a terrible error by previously in-form keeper Wildsmith saw Jota double the lead with just 21 minutes gone.
Marco Matias was harshly sent off after an altercation with Blues' Cohen Bramall and just before the break, Jota made it 3-0.
Lucas Joao looked like making a game of it when he pulled one back with a fine left foot strike nine minutes into the second half.
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That was Wednesday's first league goal since Boxing Day.
Daniel Pudil then became the second Owls player to receive a red card, with his second caution coming after an apparent trip on the acrobatic Maxime Colin.
The first was also harsh as he was booked for trying to retrieve the ball from Blues keeper David Stockdale was attempting to hold up play in the first half.
Luhukay said it was a 'disappointing day' where anything that could go wrong, did so.
"It was a disappointing day for us," he said. "It goes wrong, whatever can go wrong today in the first half.
"We went two goals behind where we made very big mistakes for the first and second goal.
"After that we also had the red card. I don't know what the referee has seen from that situation. From my position it was not a hard moment between two players but the referee decided for the red card so for us, it was very hard.
"Then, before half-time, they get the third goal, so it was mentally not so easy to come back in the game. At half-time we spoke and said that we must stay together.
"We came back to try in the last 45 minutes and we made it 3-1. Then after we had the second red card so what went wrong today also happened in the second half."
Wednesday haven't won in the Championship since that victory over Nottingham Forest on December 26 and are seven points off the relegation zone.