CBP officers seized the drugs and vehicle.
U.S. Border Patrol said on Friday its resources were being stretched thin by larger and larger groups of Central American families left by smugglers in remote locations along the U.S. Mexico border.
CBP said it would release more information about the shooting.
According to Nogales Mayor Arturo Garino, a man was attempting to cross the border from the United States into Mexico when a border patrol agent asked him to stop.
Migrants told authorities that buses and trucks dropped them off throughout the night on a nearby Mexican highway that runs parallel to the border and they entered the USA together to wait for agents to find them.
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The resolution-which says such a wall "is only found in a war, prison or battle setting" and has no place in the city-says that if the government does not remove the wire, it will file a lawsuit to have it taken down.
The resolution characterized the installation of the razor wire, recently installed by U.S. Army personnel, as "not only irresponsible but inhuman".
After shots were fired, CBP says the auto continued to roll across the border when Mexican authorities gave the man first aid.
Nogales, a city of about 20,000 people, is largely reliant on Mexican shoppers and cross-border trade.
He said he was shocked to see coils of concertina wire being hung from top to bottom on the Arizona side of the border in nearby Nogales, adding it makes no sense from a security standpoint. Illegal crossings in that area have dropped steeply in recent years.