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: "Football sometimes you do things that 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.
"He knows what he's done and I'm not going to kill him, you try to make sure he controls himself better the next time."