A goal in each half from Marco Matias and Adam Reach ensured the Owls prevailed at the Stadium of Light to set up a clash with Wolves in the week beginning August 27.
Luhukay, who made seven changes to his starting line-up, told The Star: "I'm happy to have a home draw. You don't have the travel to think about.
"We will play at home and try to give our best.
"We played a fantastic game at Wolves last year.
"They are the Premier League club so they are the favourites."
The Dutchman handed first-team debuts to Alex Hunt, who was taken off at half-time for tactical reasons, Matt Penney and Jack Lee.
Wednesday were made to work hard in the first half but raised their game after the restart.
"It is of course satisfying when you win and get through to the next round," admitted Luhukay, who could not shed any light on whether the club's transfer embargo is close to being lifted. "It is positive for confidence and trust.
"It was a tough game for us. We had a lot of problems in the first half.
"We had to defend well. Sunderland created quite a few chances.
"We changed the formation in the second half from 3-5-2 to 4-3-3 and we had more control.
"I think the team did a good job in the second half. We were better defensively.
"The dangerous situation was more from set pieces from Sunderland in the second half and we made a very good second goal from Adam Reach and that was decisive.
The only negative for the Owls was defender Joost van Aken picking up an ankle injury just after the hour mark.
"He has problems on his ankle and we will see tomorrow (Friday) how it is. He may have a better feeling or it might be worse.
"We must wait to see how his ankle is and whether it is swollen or not."