Douglas L. Wright, 26, Brandon L. Baxter, 20, and Anthony Hayne, 35, were arrested Monday evening by the FBI's Joint Terrorism Task Force, according to the statement.
They have been charged with conspiracy and attempted use of explosive materials to damage physical property affecting interstate commerce.
Connor C. Stevens, 20, and Joshua S. Stafford, 23, were also arrested.
"The public was never in danger from the explosive devices" because an undercover FBI agent was involved and the explosives were inert, the bureau said in a news release.
"The defendants were closely monitored by law enforcement," it said.
"Clearly this was a sting operation," said Fran Townsend, a CNN National Security contributor, describing the use of at least one covert agent in the operation.
Authorities said three of the men are self-proclaimed anarchists who had considered "a series of evolving plots over several months."
The men allegedly conspired to acquire C-4 explosives and build two improvised explosive devices to be remotely detonated, according to the FBI.
Their initial plan, officials said, was to use smoke grenades to distract local law enforcement in an effort to "topple financial institution signs atop high rise buildings."
They allegedly discussed several plots, eventually settling on the Brecksville-Northfield High Level Bridge, which spans the Cuyahoga Valley National Park and carries a four-lane highway.