Freiburg debutant Yoric Ravet was dismissed after a Video Assistant Referee review just before the half-hour mark for a reckless challenge on Marcel Schmelzer, referee Benjamin Cortus having initially brandished a yellow card.
From then on, Dortmund poured men forward and ended the match with 83 per cent possession, but it was not enough as Freiburg clung on to a commendable point.
And Bosz was thoroughly unimpressed with his side for failing to make the most of their superiority.
"I'm disappointed," Bosz told reporters. "When you play for an hour with a man extra, you must win. There is enough time to score.
"To leave with a draw isn't enough. That was not enough today. For us, there are two lost points."
Schmelzer had to be withdrawn as a result of Ravet's tackle and midfielder Gonzalo Castro blamed referee Cortus for setting the tone by failing to strictly punish earlier challenges.
"This is the problem: if you, as a referee, do not give an early sign with a yellow card then something happens, like this foul, and one of us is hurt again," he said.