"There was a single shooter that was shot multiple times at the scene. He was not killed as previously reported. He is currently in custody and in stable condition," Cone said.
Hasan was born in the US to Muslim Palestinian parents who had emigrated from a small town near Jerusalem, US media said.
Further bloodshed was narrowly prevented when Hasan was apparently blocked from reaching a graduation ceremony attended by some 600 people, just metres away from the scene.
"Thanks to the quick reaction of several soldiers, they were able to close off the doors to that auditorium," Cone said.
Josh Rushing, Al Jazeera's correspondent at Fort Hood, Texas, said: "[Hasan] is a first-generation American. He joined the army after high school and went to the Virginia Tech university to get a psychiatry degree through a military programme.
"He became a psychiatrist at the Walter Reed Army Medical Centre in Washington ... where he counselled soldiers coming back from war.
"Every day, he heard how horrible those stories were and he really started to question the wars, according to what his cousin and sources who knew him said.
"Hasan became more devout in his religion and started arguing with soldiers about whether the wars were right or not, to the point where he received disciplinary action and negative work reviews.
"He was transferred to the medical facility here at Fort Hood, where apparently these feelings continued.
"It raises a major question - how can a person responsible for the mental health of soldiers returning [from war] be allowed to continue in this profession when he has these kinds of questions himself?"
The rampage occurred at a time of stress for the US armed services burdened by two wars, with commanders struggling to ease the effect of repeated combat tours on troops and their families.
Barack Obama, the US president, on Friday ordered flags at federal buildings to fly at half-staff until Veterans Day on November 11.
"This is a modest tribute to those who lost their lives even as many were preparing to risk their lives for their country," he said.
"We don't know all of the answers yet. I would caution against jumping to conclusions until we have all of the facts."
Suicides in the army hit a record level last year, with at least 128 taking their lives, and are on track to set a new high this year - surpassing the rate among the wider civilian population.
US commanders believe repeated deployments to Iraq and Afghanistan have played a role in the spike in suicides, as well a surge in post-traumatic stress and depression.
Hasan faced his own imminent deployment for military service, officials said.
Nader Hasan, a cousin, said Hasan was "mortified by the idea of having to deploy" and that he had been harassed by other soldiers for being a Muslim.
He told the New York Times newspaper that Hasan had retained a lawyer and sought to get out of the army before the end of his contract.
Fort Hood was locked down after the attack, which occurred on the same day as a graduation ceremony was due to go ahead at the facility.
April 3, 2009: Jiverly Wong, a Vietnamese immigrant, opens fire at an immigrant community centre in Binghamton, New York, killing 11 immigrants and two workers. Wong killed himself at the scene
March 10, 2009: Michael McLendon, 28, killed 10 people, including his
Mother and four other family members in Alabama before himself committing suicide.
February 14, 2008: Former student Steven Kazmierczak, 27, kills five students and wound 18 more in shooting at Northern Illinois University in DeKalb. He then killed himself.
December 5, 2007: Robert A. Hawkins, 19, opens fire in a shopping mall in Omaha, Nebraska killing eight people before taking his own life.
April 16, 2007: Cho Seung-Hui, 23, kills 32 students and staff at Virginia Tech before killing himself in the deadliest mass shooting in modern US history.
Some other bases across the US also stepped up security in the wake of the shootings.
Barack Obama, the US president, called the attack a "horrific outburst of violence".
"It is horrifying that they should come under fire at an army base on American soil," he said in Washington.
Patty Culhane, Al Jazeera's correspondent in Washington, said that the attack had come in spite of rigorous security protocols at the base.
"At every US military base since September 11, they do something that is called '100 per cent ID check' - that means that when you pull up to the gate, there are armed soldiers and also contractors there, and you have to have a special sticker in your car," she said.
"You also have to have a military ID. If you do not have that, you have to pull over to the side, and your car is usually swept for explosives, and you need an escort.
"Family members [of service personnel] do have ID, so they are allowed to go on to the base."
Fort Hood is home to about 50,000 troops, although Kay Bailey Hutchison, a Texas senator, said there were about 35,000 troops at the base at the time.
"Our dedicated military personnel have sacrificed so much in service to our country, and it sickens me that the men and women of Fort Hood have been subjected to this senseless, random violence," she said.