Boss Steven Schumacher praised his side’s comeback qualities but admitted the Whites made too many mistakes to save their Carabao Cup hopes.
Despite being down to 10 men inside half an hour following the dismissal of defender Sam Inwood, Wanderers three times fought back to take their first-round tie to penalties.
A superb injury-time strike from substitute Amario Cozier-Duberry made it 3-3 but a young Wednesday side shrugged off the set-back to advance on spot-kicks and set up a second-round tie at home to Leeds United.
“It was a bit of a mad game and a bit chaotic at times,” admitted Schumacher who, like the Owls, made 10 changes to his starting line-up.
“From the start it was 100mph, a bit scruffy, probably with a lot of the changes and lads who've not played loads of minutes and trying to get them up to a rhythm.
“So it was a bit mad and a few too many mistakes. There were too many errors that led to the goals and led to the big moments for them. That's something that we need to tidy up on, if we can.
“But then unbelievable effort, unbelievable character, spirit, togetherness to stay in it with 10 men. And some really good quality as well. We've had the most shots on target that we've all season with 10 men. So we've shown good character but, unfortunately, we lose it, which is disappointing.”
Jordi Osei-Tutu was the only one of the Wanderers’ side to start again on the back of last Saturday’s opening League One win and it was the full-back who nodded Wanderers’ first equaliser before half-time.
Schumacher made changes from the bench in the second half and wingers Thierry Gale and Cozier-Duberry again showed what they will bring to the Whites with classy strikes to level it up again.
But having seen the Championship side then advance 4-2 in the shoot-out, Wanderers must hope the added game time for more of their squad will pay off in the next few weeks of their League One campaign.
“At this stage of the season you can see there's players that are not quite up to full speed because it's so early. We've only played two games,” added Schumacher.
“To get an opportunity to play 90 minutes and to play 90 minutes when you're stretched and you're a man down will help the fitness levels. That was one of our main ideas and what we wanted to do.
“We also had a tough game on Saturday against Plymouth so we have to protect certain players because they’re important to us. Eoin Toal wasn't in the squad and Amario was a little bit stiff.
“We have to be mindful of that and we've got a huge game at the weekend and we want to make sure that we're fresh for that. There are things to learn and, most importantly, it looks like there's been no injuries, which is good.
“When we get back in tomorrow we’ll prepare for Barnsley and that's now right at the front of our minds and we get ready to go for a real tough game.”
Watch the full post-match interview with the Whites' Head Coach on Wanderers TV