Mitchell began the race for the United States, who went through to Friday's final as the second fastest team despite Mitchell's injury.
"As soon as I took the first step past the 200m mark, I felt it break." Manteo told the USA Track and Field website.
"I didn't want to let the three guys or the team down, so I just ran on it."
Mitchell still managed to finish the opening lap in 46.1 seconds as the US team, also featuring Joshua Mance, Tony McQuay and Bryshon Nellum, went on to set a qualifying time of two minutes, 58.87 seconds.
"It hurt so bad," the 25-year-old added. "I'm pretty amazed that I still split [close to] 45 seconds on a broken leg."
USA Track and Field chief executive Max Siegel said: "Manteo has become an inspiration and a hero for his team-mates.
"Without his courage and determination to finish, Team USA would not be at the starting line in the final. The team has rallied around him and we are all looking forward to the final days of competition."
The US men's team are the defending champions and have won every 4x400m Olympic title that they have contested.