President Barack Obama described it as "a horrific outburst of violence".
Speaking at a press conference in Washington, he said: "It is difficult enough when we lose these brave men and women abroad, but it is horrifying that they should come under fire at an army base on US soil."
He extended his condolences to the families of the victims, adding: "We will make sure that we get answers to every single question about this horrible incident."
Eleven victims were initially reported dead, but two of the injured later died, bringing the death toll to 13.
The gunman is now said to be wounded after being shot four times, but is in a stable condition in custody.
"His death is not imminent," said Lt Gen Cone.
Barack Obama: 'A horrific outburst of violence'
An official said authorities initially thought one of the victims was the shooter, causing a delay in identifying Maj Hasan as the suspect, AP reported.
Maj Hasan is a military psychiatrist and was reportedly due to be sent on a mission to Iraq or Afghanistan.
His cousin said Maj Hasan had been resisting such a deployment.
"He hired a military attorney to try to have the issue resolved, pay back the government, to get out of the military. He was at the end of trying everything," Nader Hasan told Fox News.
He also said that Nidal Malik Hasan had been battling racial harassment because of his "Middle Eastern ethnicity".
Prior to Fort Hood, Maj Hasan served as a psychiatrist at the Walter Reed Army Medical Center in Washington, which treats wounded troops from combat zones.
SHOOTINGS AT FORT HOOD US MILITARY BASE
Shooting starts 1330 local time (1930 GMT) at Soldier Readiness Processing Center near Howze Theater where graduation ceremony is due to take place
Thirteen people are killed and 30 injured
Soldiers close off Howze Theater to protect about 600 people inside
Gunman Major Nidal Malik Hasan is shot and detained
Built in 1942, Fort Hood is the largest US base in the world, home to about 40,000 personnel and includes two museums and a lake. It has been a focus for anti-war protesters
The shooting had begun at about 1330 (1930 GMT) on Thursday at a personnel and medical centre at Fort Hood, where soldiers who are preparing to deploy go for last-minute medical check-ups, Lt Gen Cone said.
He said the gunman had two weapons, one semi-automatic, which "might explain the rate of fire".
Asked whether the shootings were a terrorist act, Lt Gen Cone said: "I couldn't rule that out but I'm telling you that right now, the evidence does not suggest that."
Two more suspects were apprehended in an adjacent facility, he said, but eyewitness accounts suggesting there might have been more than one gunman were later discounted.
A serviceman stationed at Fort Hood, who asked to remain anonymous, told the BBC: "I heard the emergency announcement over the speakers outside and saw people rushing to get indoors."
Local congressman John Carter, speaking to NBC News, said gunfire had erupted half an hour before a graduation ceremony was due to begin.
'Like a city'
Fort Hood, near the town of Killeen, is the largest US base in the world.
Home to about 40,000 US troops, the base lies between Austin and Waco, about 60 miles (100 km) from each city.
Soldiers at Fort Hood are among those deployed in Iraq and Afghanistan, and some will have returned from there.
The base has a centre that deals with combat stress.
Hilary Shine, of the Killeen Fire Department told the BBC's News Channel Fort Hood was like a small city.
"It has schools, a hospital, a convenience store even. And it has a large daytime population - including civilians working on the base - with as many as 80,000 in this area during the daytime."
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