The charity in question denied raising money for weapons or ammunition, saying it was focused on humanitarian aid and disaster relief.
The FSB is thought to have discovered the transaction on her phone.
In exchange, the US freed Arthur Petrov, a dual German-Russian citizen arrested in Cyprus in 2023. He was accused of illegally exporting microelectronics to Russia for manufacturers working with the Russian military.
Russian TASS news agency said President Vladimir Putin had pardoned Ms Karelina.
The prisoner swap took place in Abu Dhabi in the early hours of Thursday.
During a Thursday cabinet meeting, Trump said the release of Ms Karelina was a request from Ultimate Fighting Championship CEO Dana White.
"Dana White, called me and he said it's the friend or the relationship of one of the fighters, UFC or one of the fighters, and Dana is an incredible guy, and we spoke to President Putin about it, and they made a deal," he said.
US Secretary of State Marco Rubio confirmed Ms Karelina was "on a plane back home to the United States".
He added she had been "wrongfully detained by Russia for over a year".
"President Trump secured her release. [The President] will continue to work for the release of ALL Americans."
CIA director John Ratcliffe was present at the exchange, the Wall Street Journal said.
Ms Karelina's parents thanked Trump and Putin.
"I guess that deal must have involved them both," her father, Pavel, told the Wall Street Journal. "We are beside ourselves with happiness.
"The first seconds of our chat were all pure emotions, I can't even remember what we were saying, it was like one explosion of happiness."
It is the second prisoner swap between Russia and the US in less than two months.
In February, Russian national Alexander Vinnik – who was imprisoned in a US jail on money laundering charges – was freed in exchange for the release of American schoolteacher Marc Fogel.
The exchange comes as Moscow and Washington try to improve their relations.
US and Russian officials met in Istanbul on Thursday to hold another round of talks aimed at restoring some of the embassy operations that were scaled back following the Ukraine invasion.