The incident began at about 9.15am local time (7.15pm AEST) on Thursday, when the suspect unsuccessfully tried to enter the FBI's visitor screening facility. He then fled the area in a white vehicle, the FBI said.
About 20 minutes later, a state trooper spotted the vehicle in nearby Warren County and attempted a traffic stop.
A chase ensued as the suspect fired his weapon at the state trooper, Ohio State Highway Patrol spokesman Nathan Dennis said during a news briefing.
The vehicle eventually came to a stop and gunfire was exchanged between the suspect and police officers, he said. No police were injured, Dennis said.
The hours long standoff ended later in the afternoon, Clinton County Emergency Management Agency said, with the FBI later confirming the suspect had died.
Several highways in the Cincinnati area were closed during the standoff.
"Law enforcement has traded shots with a male suspect who is wearing a grey shirt and body armour," Clinton County Emergency Management Agency said earlier, urging people in the area to stay inside and lock their doors.
NBC News reported the suspect was armed with an AR-15-style semiautomatic rifle and fired a nail gun into the FBI building.
Further details and the suspect's motive were unclear.
The FBI has been the subject of online threats since its agents searched former president Donald Trump's Florida estate this week as part of an investigation into documents removed from the White House as he was leaving office in January 2021.
