The 25-year-old ran the opening leg of the race and still managed a 46.1s lap before handing over to his teammate.
"As soon as I took the first step past the 200m mark, I felt it break," the 25-year-old 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.
"It hurt so bad. I'm pretty amazed that I still split [close to] 45 seconds on a broken leg."
USA Track and Field chief executive Max Siegel described Mitchell as an "inspiration" and "hero" after his efforts, which meant that the US quartet could progress to the final of the event.