In a statement, the Holt family said, "We thank you for your collaboration during this time of anguish".
In televised comments earlier this month, the No. 2 official in Venezuela's ruling Socialist Party, Diosdado Cabello, described Holt as "the head of USA espionage in Latin America" and said that he would remain behind bars.
Authorities arrested him on June 30, 2016, and accused him of stockpiling an assault rifle and grenades, suggesting his case was linked to other unspecified USA attempts to undermine Maduro's rule amid deep economic and political turbulence.
At a news conference in Caracas earlier on Saturday, Venezuelan Communications Minister Jorge Rodriguez said Holt and his wife were freed as part of efforts by Maduro's government to maintain "respectful diplomatic relations" with Washington.
The wife of the American man held for two years in jail in Venezuela says they anxious until their plane was in the air that their release and flight to the US would somehow fall apart.
Hatch announced Holt's release on Friday, characterizing it as the end result of two years of negotiations with Maduro's government. "Senator Hatch has worked tirelessly on the Holt family's behalf, and I was honored to play a small role in bringing Josh and his wife home to the United States". "We have others coming", he said, singling out pastor Andrew Brunson, held by Turkey since an attempted coup in 2016.
Hatch praised the "pivotal efforts" of Corker, chairman of the Senate Foreign Relations committee, to win Holt and Caleno's freedom.
Until Trump's tweet on Saturday, the US had stopped short of publicly calling Holt a "hostage". Restatements of demands for his release were issued by senior U.S. politicians, and Todd Robinson, then the chief of mission at the U.S. Embassy in Caracas, showed up at Venezuela's Foreign Affairs Ministry in Caracas to demand information about his safety.
"I'm just overwhelmed with gratitude", Holt told Trump.
After being sworn in for a new six-year term this week - in a vote condemned by the U.S.as a "sham" after the main opposition parties boycotted it - Maduro called for the release of some detained prisoners, without naming anyone specifically. "They want to kill me and paint the walls with my blood".
'This wouldn't have happened without you, ' he said.
Holt went to Venezuela to marry a woman he met online, but was arrested, on a claim he was a trained gunman sent to undermine President Nicolas Maduro.
Earlier, NSC officials told CNN's Boris Sanchez that the U.S. "offered nothing" to Venezuelan leadership in exchange for the release of Holt. (Holt family photo via AP) In this image provided by the Holt family, Joshua Holt, his wife Thamara and her daughter Marian Leal, board a plane at the airport in Caracas, Venezuela, Saturday, May 26, 2018. Relations between the United States and Venezuela have been strained recently, particularly over the election which last week returned Maduro to power. The Venezuelan government released 20 activists from prison the same day.