Griner arrived in the UAE capital of Abu Dhabi by private plane from Moscow as Bout was flown in on a private plane from Washington.
Bout, nicknamed the "merchant of death," embraced a Russian official who greeted him and smiled broadly as he was led away.
A joint UAE-Saudi statement said the UAE president and Saudi crown prince led mediation efforts that secured Griner's release.
Griner was exchanged for Bout, a onetime Russian weapons dealer who had been convicted in the United States and imprisoned for 10 years.
During the prisoner swap, Griner was met on the tarmac in UAE by chief U.S. hostage negotiator Roger Carstens.