Former Denmark youth international Madsen was forced off after just 30 minutes on Tuesday, with Saints already one goal ahead through Finn Azaz.
The home side were frustrated as Stephan demanded the midfielder sit down just before exiting the playing area, in order to force a lengthy stoppage.
But despite the last-ditch team talk, QPR lost the momentum they had been building before Madsen’s injury, and Saints scored again before half-time.
Saints head coach Tonda Eckert admitted Madsen’s withdrawal changed the game, with the central midfielder causing Saints problems in wide areas.
Stephan said: “Tough evening. We conceded the goals at the bad moments, at the beginning of the first half and the last second of the first half.
QPR boss Julien Stephan and Saints head coach Tonda Eckert (Image: PA)
“Then again, after two minutes or three minutes at the beginning of the second half. But the crucial moment came when we were in a good moment.
“When Nico had his injury, it was a crucial moment for us, because during the 15 minutes after we conceded the first goal, we had a good reaction.
“It was a very good collective reaction, without the ball, with the ball. I felt on the bench that it was possible to come back and to create something.
“But when we lost Nico, it changed the momentum completely. We suffered in the last 10 minutes of the first half, and we conceded at the last second.
“Afterwards, it was more a question in the second half of management, to think about the next one, how we could give some minutes to some players.
“We were thinking how we could give some rest to the other players, and think about the next one. So I want to take full responsibility.
“I make the choice, and this is my responsibility. I apologise to the fans who came to the stadium and who stayed behind their television today.
“It’s not the real face of QPR.”