"That's why they have the random southbound checks", Nogales Mayor Arturo Garino said.
The officer fired his gun at the vehicle after it fled south into Mexico eventually stopping several yards from the port, the release said.
The border crossing was initially closed after the shooting, but later reopened.
The council's resolution says the razor wire would harm or kill anyone who scales the wall and "is only found in a war, prison or battle setting" and shouldn't be in downtown Nogales.
On Wednesday, Nogales City Council passed a resolution that would sue the federal government if the barbed wire was not removed.
Officers referred the tractor-trailer to a secondary inspection station, according to information obtained from CBP officials.
Agents of El Paso Sector U.S. Border Patrol conduct a Mobile Field Force training exercise in the Anapra area of Sunland Park, New Mexico, as seen from the Mexican side of the border in Ciudad Juarez, Mexico January 31, 2019.
The vote comes after USA military troops installed additional layers of wire along the fence over the weekend, and the day after President Donald Trump highlighted border security in his State of the Union address.
Numerous migrants who may seek passage with smugglers in their journey through Mexico cross the border and turn themselves into USA authorities to seek asylum in the United States, a drawn-out court process that can take months or years to resolve. The AP reports illegal crossings in that area have dropped steeply in recent years.
Authorities closed the DeConcini Port of Entry near Nogales after a man was shot attempting to run over a border patrol agent on Thursday night.