Luhukay, appointed late last night, saw Wednesday create the better chances but their finishing let them down. Atdhe Nuhiu and Lucas Joao both hit the woodwork as the Owls' winless run stretched to three matches.
The result means the Champinship club have recorded just one victory in their last 11 outings.
Interim boss Lee Bullen, in his final match in charge, made three changes to the team who were comprehensively turned over by Burton Albion on New Year's Day.
He handed first-team recalls to Liam Palmer, George Boyd and Atdhe Nuhiu. It was Boyd's first start since September after recovering from shoulder surgery. Jack Hunt, Jacob Butterfield and Jordan Rhodes dropped.
Sam Hutchinson was left out of the matchday squad while youngsters Ash Baker and Connor O'Grady.
Wednesday, missing a raft of players through injury, almost broke the deadlock inside the opening minute. Liam Palmer whipped in a superb cross from the right which Nuhiu met with a powerful header that rattled the crossbar.
For our Owls news and opinion on twitter follow hereThe visitors dominated possession in the early stages and Adam Reach fired over from just outside the penalty area after League Two Carlisle only partially cleared a David Jones free-kick.
Nuhiu spurned another glorious chance to put the Owls ahead in the 11th minute. The striker wriggled his way into the penalty area and saw his fierce shot pushed to safety by Jack Bonham.
It took the Cumbrians until the quarter of an hour mark to threaten, Hallam Hope's left foot drive from an acute angle was well held by Joe Wildsmith.
Wednesday carried the greater attacking threat and Jones' rasping left foot drive forced Bonham into action.
The closest Carlisle came to opening the scoring in a first half packed full of endeavour but low on quality was in the 26th minute when right wing back James Brown slipped in Hope but the former Owls loanee drilled a shot over.
For more news from Hillsborough click hereLone striker Sam Cosgrove received the first yellow card of the afternoon after a late lunge on Morgan Fox.
It was Carlisle who created the first meaningful opportunity after half-time when Reggie Lambe tried his luck from long range but his shot went high over the top.
Moments later, Joao narrowly glanced over Palmer's delightful cross as Wednesday pressed hard for the opener.
Joao then combined well with Morgan Fox to feed Boyd, who was denied by Bonham.
Portugal international Joao was unlucky not put the Owls in front just before hour mark, seeing his header from Jones' free-kick hit the post.
Wednesday piled on the pressure and Wallace cut inside on to his left foot and fizzed a fierce shot inches wide.
The hosts then thought they had taken the lead when Sam Cosgrove headed home Jamie Devitt's cross but he was adjudged to have strayed into an offside position.
Bullen threw on Jordan Rhodes and Marco Matias in search of a winner. Rhodes nodded off target after good play by Wallace and Palmer as Carlisle held on.
Carlisle: Bonham; Liddle, Hill, Parkes; Brown, Grainger, Jones, Devitt; Lambe (O'Sullivan 86); Hope (Miller 78), Cosgrove (Bennett 78). Substitutes: Bacon, T Miller, Nabi, Ellis.
Owls: Wildsmith; Palmer, Loovens, Venancio, Fox; Jones; Wallace, Boyd (Rhodes 68); Reach; Joao, Nuhiu (Matias 79). Substitutes: Dawson, O'Grady, Butterfield, Baker, Pudil.
Attendance: 7,793 (1,816)