The Owls lost captain Glenn Loovens to a red card after just four minutes at Pride Park, conceding a penalty which was converted by Matej Vydra.
The visitors also had two penalty appeals of their own turned down.
Craig Forsyth pushed Steven Fletcher, with the Owls striker one-on-one with goalkeeper Scott Carson, and Curtis Davis lunged in on Gary Hooper.
The Rams wrapped up victory with a late goal from Bradley Johnson, but Carvalhal – furious at the officials – remains convinced the Owls will earn promotion.
“The gap is increasing but I still believe we can get promoted,” said Carvalhal. “I have a feeling we will get promoted.
“The way that my players are training and the way that my players are focused on the work make me believe that we will turn things and we will start winning.
“Sometimes when you have a very bad situation, you come back more strong and focused and I believe our team will provide a reaction and be in a higher position at the end.”
Defeat at Derby means Wednesday have picked up just four points out of a possible 18, but Carvalhal was impressed with his 10-man Owls.
“I’m working hard,” insisted Carvalhal, who talked with owner Dejphon Chansiri pitch-side before kick-off. “If my players didn’t run, fight and have quality, attitude and soul, I would be worried. I saw my players running, fighting and creating chances.
“They played with a fantastic attitude from the beginning to the end. To me, it’s very clear we are on the way to winning games.
“I don’t see the arms down and my players having no soul and not a good attitude to play. These kind of things make me more strong for the next games.
“If we keep on showing the attitude and quality we produced against Derby, the victories will come.”
Wednesday are five points off the play-offs and face crucial back-to-back games at Hillsborough against Barnsley on Saturday and Millwall four days later.
Carvalhal said: “We must start to win games. We have to continue our work with determination and focus now on the next two home games.
“The most important thing is that we focus on Barnsley. It is a derby.
“We know we are not in a good position. We are absolutely determined to change things.”
Match report: Page 5