Police identified Gerald Jim, 29, as the suspect, Farmington Police Lt. Taft Tracy said Monday.
Jim was taken to San Juan Regional Medical Center after the crash.
He was still being treated at the hospital on Monday and he had not been charged with a crime, Tracy said.
When he is released from the hospital, he will be arrested on suspicion of DWI, aggravated fleeing of law enforcement and driving with a suspended license, Tracy said. If convicted, it would be Jim's fifth DWI.
Kathryn Pettijohn, a hospital spokeswoman, said Monday evening that Jim was in fair condition.
A resident called police at about 5 p.m. Saturday evening and reported a driver was swerving on Main Street and Piñon Hills Boulevard, said Farmington police Sgt. Baric Crum. A police officer saw the suspect nearly collide with another car in the area of Main Street and 30th Street.
When the officer tried to pull the suspect over the person increased his speed and fled to Cliffside Drive, Crum said.
The driver lost control and the car ended up pinned against a tree on the north side of the road.
Firefighters stabilized the car, took off the front windshield and removed Jim from the vehicle, Crum said.
Police also said Monday that no arrests have been made in the stabbing of a 24-year-old man on Saturday.
On Saturday afternoon, a stabbing was reported to police when an injured person in a white Chevy pickup truck left the vehicle in the middle of Browning Parkway and was transported to the hospital by a passerby to be treated for minor stab wounds on his left arm.
Police responded to the hospital and the vehicle. The truck was parked in the street for about an hour on Saturday while police investigated.