Striker Barnes was dismissed for a late two-footed tackle on Owls midfielder Kieran Lee in the 32nd minute.
Leroy Lita volleyed Wednesday in front after five minutes for his second goal in two games and fellow loanee Danny Pugh doubled the lead from the free-kick which followed Barnes' dismissal.
Brighton midfielder Andrea Orlandi halved the deficit nine minutes after the restart following Lewis Buxton's poor clearance but Wednesday winger Michail Antonio prodded in a Pugh corner moments later to restore the hosts' two-goal advantage.
Wednesday boss Dave Jones and Brighton first team coach Charlie Oatway were later sent to the stand following a second-half touchline fracas.
Poyet said of the red card: "In football, sometimes you do things that (you) regret after, but (I have) nothing to complain about. I think you would normally probably say you're not happy with him (Barnes).
"Angry? It's difficult to say because I played football, I know when you're on the pitch you do things which sometimes you regret after but I was one of those players so I'm not going to say anything to Ashley Barnes - I hope it doesn't happen too much."
Wednesday manager Jones dismissed the touchline incident as 'handbags' and said his side fully deserved the victory.
"Things happen within a game, the sending off was pivotal in it as well - and deservedly so, the lad stamped on him," said Jones. "Then I just felt every challenge we made they (Brighton) were trying to even it up and I didn't think that was right and that's what the fracas was over. It's handbags at the end of the day, it's done and dusted now, it's over.
"I think the referee did the right thing just to calm it all down because it just would've just carried on at the end of the day. I'll see what he puts in his report and if he gets it right."