They remain too close for comfort to the drop zone after suffering a fifth defeat in 13 games under the Dutchman at Ashton Gate.
Some Wednesday fans even took to Twitter demanding Luhukay be given the sack and the Owls’ chief admitted: “We now have two home games (Bolton follow) and we must be much better. In the next few days we must give the players confidence to get a good result on Tuesday night.”
Luhukay refused to be critical of his players after a crushing defeat that leaves Wednesday seven points ahead of the bottom three with 11 games to play.
“They were too strong for us and had too much quality. You have to give them a lot of credit for the way they played,” said Luhukay.
“Their two strikers caused us a lot of problems and we couldn’t handle them in offensive situations. One of them is among the top scorers in our league.
“Sometimes you have to admit that an opponent is stronger than yourselves and that was the case. In the first half we didn’t only struggle at the back. When we got possession we had no solution as to how to use it.
“At half-time we tried to put some confidence back into the players and the second half was better from out point of view. But when you are 3-0 up, as they were, you can control the game and they did that well.”
Bobby Reid’s first hat-trick and an exhilarating team performance had Bristol City head coach Lee Johnson beaming.
The win, which also featured a goal from Josh Brownhill, took the Robins back into the play-off places.
But it was the manner of the victory that most delighted Johnson. “Some of our football was very impressive and the game was everything I could have hoped for,” said the former Barnsley manager.
“I wanted us to get our mojo back, by which I meant the confidence on the ball to play out from the back and produce the football we pride ourselves on.”
The hosts took the lead on 13 minutes when Reid played a one-two with Brownhill on the edge of the box and ran through unchallenged to beat Joe Wildsmith. It was 2-0 on 35 minutes as Reid exchanged passes with Marlon Pack before firing past Wildsmith.
Wednesday’s fragile defence was breached again two minutes before the break, Jamie Paterson’s clever flick putting Brownhill in to shoot past the unprotected Wildsmith.
Reid completed his treble from the spot on 62 minutes, sending Wildsmith the wrong way after Joel Pelupessy had needlessly tripped Pack.
Bristol City: Fielding, Pisano (Kelly 64), Flint, Magnusson, Bryan, Brownhill, Smith, Pack, Paterson, Reid (Diony 77), Diedhiou (Walsh 70). Unused substitutes: Steele, Eliasson, O’Neil, Kent.
Sheffield Wednesday: Wildsmith, Palmer, Frederico Venancio, Pudil (Lees 46), Fox, Clare (Abdi 78), Pelupessy, Reach, Stobbs, Nuhiu, Lucas Joao (Boyd 74). Unused substitutes: Jones, Rhodes, Loovens, Dawson.
Referee: T Harrington (Cleveland).