After killing his father, Griego said he reloaded the guns so "he could drive to a populated area to murder more people," according to the statement.
His plan, the statement said, was to "shoot people at random and eventually be killed while exchanging gunfire with law enforcement."
Detectives spent two days collecting evidence and trying to piece together what led to the attacks, which occurred on the same day thousands of gun advocates gathered peacefully at state capitals around the country to rally against stricter limits on firearms. The "Guns Across America" events were being held just after President Barack Obama unveiled a sweeping package of federal gun-control proposals.
"There's no other way to say it, except that we have a horrific crime scene down there that we are working on," Bernalillo County Sheriff Dan Houston said Sunday.
The detectives had finished their work at the home by Monday afternoon. The metal gate at the home's entrance was shut, a small bouquet of purple flowers was on the top of the gate, and at each side there were religious signs, including one that read "Jesus is the reason for the season."
The sheriff's office identified the victims as Greg Griego, 51, his wife, Sarah Griego, 40, and three of their children: a 9-year-old boy, Zephania Griego, and daughters Jael Griego, 5, and Angelina Griego, 2. All appeared to have gunshot wounds to the head.
A sheriff's detective questioned the teen Saturday night, and the details of their conversation were spelled out in the statement.
The teen allegedly told the detective that he took a .22-caliber rifle from his parents' closet around midnight Saturday and shot his mother in the head while his younger brother slept next to her.