The cadet -- the term for trainees at the US Army's elite officers' school -- was killed when the Light Medium Tactical Vehicle flipped over in a rugged forest area used for training, said West Point Superintendent Lieutenant General Darryl Williams.
The injured included 20 cadets who were apparently in the back of the truck and two regular soldiers who were in the cab, one of them driving.
They were transported to local hospitals with light injuries including facial abrasions and a broken arm, according to a medical official.
Video from media helicopters showed the vehicle completely overturned in thick forest near the West Point campus.
Williams said the cadets, all beginning their final year of training before graduating, were on a "standard training exercise" when the accident happened.
"We don't know the details of how the accident actually happened," Williams said.
"It's very rough terrain... this is part of our realistic training."
The 117-year-old West Point supplies many if not most of the top leaders of the US army, and two presidents graduated from it: Civil War hero Ulysses S. Grant and Dwight Eisenhower, who was supreme commander of Allied forces in Europe in World War II.
In France to commemorate the 75th anniversary of the World War II D-Day landing in Normandy, President Donald Trump tweeted his concern over the accident.
"So sorry to hear about the terrible accident involving our GREAT West Point Cadets. We mourn the loss of life and pray for the injured. God Bless them ALL!" Trump wrote.