Shocking defending from the visitors allowed Rovers striker Danny Graham net a hat-trick and pile the pressure on the under-fire Owls manager.
"We want Jos out" rained down from the Darwen End from the majority of the 3,500 travelling support.
But Luhukay said: “The expectations in this club are high. We are now not living in the past, and fans maybe live in the past from the success, but last season it was also a direction where it was not good enough.
“That’s why I came in January and now we sit in the same situation so we must fight against that, and we must believe that we can take us to a better period.
“Now we are in the worst period, I know that. But in my character, it is not that I will give up, but only solve why we didn’t get a good result, or why the fans aren’t happy with this.”
The Owls were missing full-back Ashley Baker due to suspension.
Liam Palmer came in at right-back, while striker Lucas Joao returned for Matt Penney.
Harrison Reed should have put Rovers in front, but unmarked at the back post, the midfielder fired his shot against the foot of the post, although goalkeeper Cameron Dawson seemed to get a vital touch.
But Blackburn were in front on 10 minutes. Dawson saved the initial shot, but Danny Graham fired home the rebound.
Wednesday offered nothing in attack in the opening 45 minutes, and were relieved to see Bradley Dack blaze over the crossbar after arriving late in the box.
Jos Luhukay brought on Fernando Forestieri and Atdhe Nuhiu at half-time, replacing Steven Fletcher and Marco Matias, after the Owls failed to muster a single effort on goal in the opening 45 minutes.
But they were indebted to goalkeeper Dawson who kept out Derrick Williams's close-range header.
Adam Reach had Wednesday's first shot at goal in the 50th minute, but David Raya palmed it wide at his near post.
But the Owls' poor defence was exposed moments later, as Rovers cut through to slide in Dack to make it 2-0.
The goal brought chants of 'We want Jos out' from the visiting supporters in the Darwen End at Ewood Park.
Lucas Joao drilled home a 25-yard effort to give the visitors brief hope, but that was abruptly ended when Graham tapped home the rebound after Dack's effort smacked the post.
Dawson had to bet alert to deny Craig Conway's long-range effort.
But with five minutes remaining, Wednesday made it a one-goal game once again.
Barry Bannan powered into the box, his shot hit the post, and as the ball rolled back across goal, Rovers keeper Raya could only nudge the ball over the line.
But, just like earlier in the game, Wednesday's hopes of building up some pressure were undone by poor defensive work.
And Graham was the beneficiary once again, tapping home for a comfortable hat-trick of tap-ins.
Forestieri limped off clutching his hamstring just to complete a miserable afternoon at Ewood Park.