DNA found on at least one of the packages helped investigators identify Sayoc as a suspect, law enforcement officials told CNN.
He has a lengthy criminal record which includes bomb threats.
Cesar Sayoc, 56, a former stripper and professional wrestler once charged with threatening to bomb an electric company for trying to shut off his lights, was taken into custody outside an auto parts store in Plantation, Florida, near Miami.
In southern Florida, FBI agents and police swarmed the area around the strip mall in Plantation, where an AFP photographer saw a van covered in blue tarpaulin loaded onto a truck by authorities before being driven away.
The white van papered with inflammatory material that was captured by local media outlets during Sayoc's arrest did in fact belong to him.
Trump was briefed on the arrest by chief of staff John Kelly, a White House official said, and the President was planning to make remarks early Friday afternoon. Records say he threatened to "throw, place, project or discharge" a destructive device.
Court records also show that Sayoc was convicted in the 1990s in Broward County, Fla. on grand theft and stolen property charges.
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A package recovered at a Manhattan postal facility on Friday was addressed to Clapper, a CNN contributor, at the network.
Cesar Sayoc allegedly placed 13 improvised explosive devices in envelopes and addressed them to political figures across the United States, prosecutors said.
Details suggested a pattern - that the items were packaged in manila envelopes, addressed to prominent Trump critics, carried U.S. postage stamps and had the same return address for Florida-based Democratic Congresswoman Debbie Wasserman Schultz.
The development came amid a coast-to-coast manhunt for the person responsible for a series of explosive devices addressed to Democrats including former President Barack Obama, former Vice-President Joe Biden and Hillary Clinton. Director Wray confirmed this and was clear that the devices in question "are not hoax devices".
Clapper, who has frequently criticized Trump, vowed to CNN Friday morning that he wouldn't be intimidated in light of the discovery.
He said other packages could still be en route to other recipients before urging people to be vigilant and report anything suspicious. The devices were being examined by technicians at the FBI's forensic lab in Quantico, Virginia.
The packages stoked nationwide tensions and fears as voters prepared to vote November 6 to determine partisan control of Congress - a campaign both major political parties have described in near-apocalyptic terms.