The Owls earned the vital points in the South Yorkshire relegation scrap thanks to Gary Madine's first strike since early December, with Reda Johnson's header giving them a two-goal cushion.
They had to endure a nervy ending, though, as Lewis Buxton saw red for two yellows while Jacob Mellis powered home an injury-time consolation for the Reds.
Flitcroft, who saw his side drop into the bottom three after the loss, said: "We have got to be constructive, the two goals were disappointing, they came when we had momentum.
"When you have a foothold you have to make sure you lock that down.
"The second goal has really disappointed me, we had to be brave. At 2-0, the momentum is with Sheffield Wednesday and I didn't want it to turn into a rout. To come here and leave with a 2-1 is the only positive we'll take.
"I have picked these players up before. There are eight more games to look forward to. The pressure on us is to stay in the league, but the pressure is on other teams to get out of it and they are just panicking and we have to take advantage.
"The players are hurting but when we leave here, we leave proud and in a positive state of mind."