Festival organisers said one of Monday's injured runners was American and the other two were Spaniards. None were in serious condition.
A 29-year-old Spaniard was gored in the knee. The two other men were gored inside the city's bullring at the end of the run, the regional government of Navarra said.
The 25-year-old from Florida was gored in his leg and the other Spaniard, 29, was gored in the groin, it added.
TV images showed one bull repeatedly tossing and butting one runner against the wooden barriers on the edge of the ring and then goring another in the back of the leg.
The 12 bulls broke into separate groups, and one of them trailed behind the rest, making the course extra unpredictable for runners trying to scamper out of their way.
Several daredevils slipped or tripped while the bulls charged down the narrow streets.
Three other runners were taken to hospital for injuries sustained in falls during the run.
Six bulls and six steers raced along the roughly 850-metre course from a holding pen to the city's bullring in three minutes and 12 seconds.
The bulls that run each morning are killed in afternoon bullfights by professional matadors.
The first goring of the festival so far this year happened on Saturday when a bull's horn stabbed a 39-year-old Spaniard in the buttocks.
Monday was the fifth of the festival's eight scheduled bull runs.
The nationally televised early morning runs are the highlight of the nine-day festival made world famous by Ernest Hemingway's 1926 novel The Sun Also Rises.
Officials called off the hugely popular event in 2020 and 2021 because of the Covid-19 pandemic, the first time the festival had been cancelled since Spain's 1936-1939 civil war.
Sixteen people have died in the bull runs since 1910. The last death occurred in 2009 when a bull gored a 27-year-old Spaniard in the neck, heart and lungs.
His parents left a bouquet along the bull run route on Sunday on the 13th anniversary of his death.