The woman was killed near Main Street and Court Avenue. Police responded to the crash shortly after 5 p.m.
Farmington police Sgt. Casey Malone said the woman lived in San Juan County but her name was not released Wednesday night because her family hadn't been notified.
Joe Gallegos, 43, was arrested shortly after the crash on suspicion of DWI, Malone said. Other charges are likely, he said.
Malone said the woman was in the road on Court Avenue trying to pick something up when she was struck.
Gallegos was allegedly driving westbound on Main Street when he turned his Ford Explorer south on Court and hit the woman, Malone said.
The woman was pinned under the vehicle. Farmington firefighters responded and used a jack to free the victim.
Fighters determined the woman was dead, Malone said.
Before emergency responders arrived to the scene, Malone said Gallegos fled on foot.
It did not appear Gallegos had passengers, he said.
Police arrested Gallegos at a nearby mobile home complex and said he appeared to be high on narcotics, Malone said.
Investigators with the Farmington Gang Unit responded to the mobile home complex.
Investigators were trying to obtain search warrants Wednesday night because they believe Gallegos may have stashed narcotics at the complex, Malone said.