The striker was sent off for a late two-footed tackle on Owls midfielder Kieran Lee in the 32nd minute.
Leroy Lita volleyed Wednesday in front after five minutes, and Danny Pugh made it 2-0. Andrea Orlandi halved the deficit nine minutes after the restart but Michail Antonio settled it for the Owls.
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 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."
"I think the red card changed the game, especially conceding the goal after that incident - one of the worst, if not the worst, goal we've conceded this year, which is disappointing.
"Then the second-half we were, for 10 to 15 minutes, absolutely outstanding with 10 until we score the goal and you're thinking: 'here we go, coming back, nerves', and then we conceded possibly the second worst goal of the season."