The Times said that Rohade and Ludin found a Pakistani army scout who took them to an army base.
They were flown to the US Bagram military base in Afghanistan on Saturday, the newspaper reported.
Afghanistan's Taliban on Sunday denied any involvement in Rohde's kidnapping.
"Even at that time when they were kidnapped we had not claimed their kidnapping," Zabihullah Mujahed, a Taliban spokesman, said.
"And now ... we are not aware of how they were freed or escaped. We're not involved at all and we do not know who had kidnapped them."
In a statement, Hillary Clinton, the US secretary of state, said she was pleased that Rohde's ordeal had ended.
"I would like to thank the governments of Pakistan and Afghanistan for their assistance in ensuring his safe return," she said.
Bill Keller, the executive editor of the Times, said the newspaper had kept quiet about the kidnapping in order to avoid increasing the danger to the men.
The newspaper said that Keller declined to discuss efforts to win the men's release, but said no ransom was paid and no Taliban or other prisoners were released.
"Kidnapping, tragically, is a flourishing industry in much of the world. As other victims have told us, discussing your strategy just offers guidance for future kidnappers," he said.
Rohde had been in Afghanistan working on a book.