CBS News Correspondent Bob Orr reports the media was kept away, but photos released by Juan Miguel's attorney seem to show a happy, emotional reunion.
Witnesses say Juan Miguel boarded the airplane and embraced Elian with a "bear hug."
The 6-year-old, who had not seen his father since November, "nuzzled down in his father's chest," according to witnesses.
As Juan Miguel carried Elian off the plane, federal agents were said to be "teary-eyed."
Sources say Elian was obviously tired, but happy.
He smiled as he played with his baby half-brother and excitedly talked about the early morning helicopter ride out of Miami that preceded his flight to Washington.
But the boy's travels are not over.
From Andrews Air Force Base, were the reunited father and son are to spend a few days, Elian and his family will be going to a secure private retreat, the Wye River Conference Center, in a remote section of Maryland's eastern shore.
It is the same place that hosted the Middle East peace talks in 1998.
It's not clear if Elian's Miami relatives will be welcome there. Later that day, some of them arrived in Washington hoping for a meeting but were turned away.
That represents a reversal of fortunes, CBS News Correspondent Russ Mitchell reports: after months of waiting, Juan Miguel Gonzalez, a 31-year-old resident of the small Cuban town of Cardenas who holds a relatively lucrative job as a cashier in a tourist resort, has Elian.
Meanwhile, the Miami relatives, for the first time in months, are the ones hoping for a chance to be with the boy.
There appears to be little love lost between the two sides of the Gonzalez family.
After Elian's miraculous rescue at sea, Miami relatives got custody of the boy without his father's consent.
Juan Miguel Gonzalez, from the outset, asked for his son to be returned. It was learned that Elian spent much of his time at his father's home, even though his mother held primary custody since the couple's divorce.
"Please unite me with my son immediately please don't abuse my son any longer. I love my son and I would like him to be at my side with me. I ask you to help me," Juan Miguel has pleaded.
But the Miami relatives made clear their desire to keep Elian in the United States, and the dispute came to a head, Gonzalez told CBS News Anchor Dan Rather in an exclusive interview for 60 Minutes, when the Miami relatives released a videotape in which Elian asked to stay in the U.S. Gonzalez said his boy was coaxed.
"I saw the betrayal inside my boy, that they'd make him do these things. Actually, he's suffering more here amongst them than he suffered at sea," the father said.
Ratheasked, "After you saw that videotape the first time, did you weep?"
"I'm going to tell you the truth. I don't have any tears left. I cried too much," Juan Miguel responded.
Miami family members claim the father's desire to bring his boy home is motivated not by love, but by communist conviction.
But in a 60 Minutes II interview, Rather asked the father, "Does Fidel Castro call the shots on you, for you and about this effort to get the child back?"
"Why do you have to mix Fidel Castro into this? It's simply me, his father asking for my son. I want to return to Cuba. I like Cuba. I love Cuba. That's where I want to live. That's where I want my son to be raised at my side," he replied.