The unidentified man from San Francisco is reportedly 46 years old, and sustained a goring injury in the neck which required emergency surgery.
Two others were taken to hospital with head injuries and a total of 48 others were treated by the Red Cross.
Since the festival began in 1910, 15 people have been killed in the run, majority due to being gored.
The running of the bulls draws about 1 million spectators from around the world to Pamplona every year.
It also involves religious processions, parties and concerts and was depicted in the 1926 Ernest Hemingway novel The Sun Also Rises.
The bulls in the inaugural run featured bulls from the Puerto de San Lorenzo cattle breeder, who supplied the bulls previous year, resulting in one goring.
On Monday at 8am, six bulls were released from the Santo Domingo corral and led by tame oxen down a 848.6-meter route through narrow, cobblestoned streets ending in the bullring, where they will face a torero in the afternoon.
One of them stumbled toward the end, causing panic and at least one goring when it charged at some racers.
The third person was a 40-year-old man from La Rioja in northern Spain, according to the hospital.
The hospital said two of the goring victims had been Americans.
Bull fights and bull runs are a traditional part of summer festivals across Spain.
The high-profile trial of a gang who became known as La Manada, "the wolf pack" connected to an assault on a young woman in 2016 has led to ongoing protests.